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<div class="section" id="fwk-redden-ch01_s01" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">1.1</span> Review of Real Numbers and Absolute Value</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch01_s01_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch01_s01_o01" numeration="arabic">
<li>Review the set of real numbers.</li>
<li>Review the real number line and notation.</li>
<li>Define the geometric and algebraic definition of absolute value.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch01_s01_s01" version="5.0" lang="en">
<h2 class="title editable block">Real Numbers</h2>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p01">Algebra is often described as the generalization of arithmetic. The systematic use of <span class="margin_term"><a class="glossterm">variables</a><span class="glossdef">Letters used to represent numbers.</span></span>, letters used to represent numbers, allows us to communicate and solve a wide variety of real-world problems. For this reason, we begin by reviewing real numbers and their operations.</p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p02">A <span class="margin_term"><a class="glossterm">set</a><span class="glossdef">Any collection of objects.</span></span> is a collection of objects, typically grouped within braces { }, where each object is called an <span class="margin_term"><a class="glossterm">element</a><span class="glossdef">An object within a set.</span></span>. When studying mathematics, we focus on special sets of numbers.</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p03"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0001" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>ℕ</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>{</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>…</mn><mo>}</mo></mrow></mrow><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>N</mi><mi>a</mi><mi>t</mi><mi>u</mi><mi>r</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>N</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>s</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>W</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>{</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>…</mn><mo>}</mo></mrow></mrow><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>W</mi><mi>h</mi><mi>o</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>N</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>s</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>ℤ</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>{</mo><mrow><mn>…</mn><mo>,</mo><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>…</mn><mo>}</mo></mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>g</mi><mi>e</mi><mi>r</mi><mi>s</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p04">The three periods (…) are called an ellipsis and indicate that the numbers continue without bound. A <span class="margin_term"><a class="glossterm">subset</a><span class="glossdef">A set consisting of elements that belong to a given set.</span></span>, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0002" display="inline"><mo>⊆</mo></math></span>, is a set consisting of elements that belong to a given set. Notice that the sets of <span class="margin_term"><a class="glossterm">natural</a><span class="glossdef">The set of counting numbers: {1, 2, 3, 4, 5, …}.</span></span> and <span class="margin_term"><a class="glossterm">whole numbers</a><span class="glossdef">The set of natural numbers combined with zero: {0, 1, 2, 3, 4, 5, …}.</span></span> are both subsets of the set of integers and we can write:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p05"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0003" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>ℕ</mi><mo>⊆</mo><mi>ℤ</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>W</mi><mo>⊆</mo><mi>ℤ</mi></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p06">A set with no elements is called the <span class="margin_term"><a class="glossterm">empty set</a><span class="glossdef">A subset with no elements, denoted Ø or { }.</span></span> and has its own special notation:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p07"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0004" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>{</mo><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>}</mo></mrow><mo>=</mo><mo>Ø</mo><mtext> </mtext></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mtext> </mtext><mi>S</mi><mi>e</mi><mi>t</mi></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p08"><span class="margin_term"><a class="glossterm">Rational numbers</a><span class="glossdef">Numbers of the form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0005" display="inline"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow></math></span>, where <em class="emphasis">a</em> and <em class="emphasis">b</em> are integers and <em class="emphasis">b</em> is nonzero.</span></span>, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0006" display="inline"><mrow><mi> </mi><mi>ℚ</mi></mrow></math></span>, are defined as any number of the form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0007" display="inline"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow></math></span> where <em class="emphasis">a</em> and <em class="emphasis">b</em> are integers and <em class="emphasis">b</em> is nonzero. We can describe this set using <span class="margin_term"><a class="glossterm">set notation</a><span class="glossdef">Notation used to describe a set using mathematical symbols.</span></span>:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p09"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0008" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>ℚ</mi><mo>=</mo><mrow><mo>{</mo><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>|</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>ℤ</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>b</mi><mo>≠</mo><mn>0</mn></mrow><mo>}</mo></mrow><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>R</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>N</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>s</mi></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p10">The vertical line | inside the braces reads, “<em class="emphasis">such that</em>” and the symbol <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0009" display="inline"><mo>∈</mo></math></span> indicates set membership and reads, “<em class="emphasis">is an element of</em>.” The notation above in its entirety reads, “<em class="emphasis">the set of all numbers <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0010" display="inline"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow></math></span> such that a and b are elements of the set of integers and b is not equal to zero.</em>” Decimals that terminate or repeat are rational. For example,</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0011" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mn>0.05</mn><mo>=</mo><mfrac><mn>5</mn><mrow><mn>100</mn></mrow></mfrac><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mn>0.</mn><mover accent="true"><mn>6</mn><mo stretchy="true">–</mo></mover><mo>=</mo><mn>0.6666</mn><mo>…</mo><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mtd></mtr></mtable><mi> </mi></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p12">The set of integers is a subset of the set of rational numbers, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0012" display="inline"><mrow><mi>ℤ</mi><mo>⊆</mo><mi>ℚ</mi></mrow></math></span>, because every integer can be expressed as a ratio of the integer and 1. In other words, any integer can be written over 1 and can be considered a rational number. For example,</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p13"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0013" display="block"><mrow><mn>7</mn><mo>=</mo><mfrac><mn>7</mn><mn>1</mn></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p14"><span class="margin_term"><a class="glossterm">Irrational numbers</a><span class="glossdef">Numbers that cannot be written as a ratio of two integers.</span></span> are defined as any numbers that cannot be written as a ratio of two integers. Nonterminating decimals that do not repeat are irrational. For example,</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p15"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0014" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi mathvariant="italic">π</mi><mo>=</mo><mn>3.14159</mn><mo>…</mo><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><msqrt><mn>2</mn></msqrt><mo>=</mo><mn>1.41421</mn><mo>…</mo></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p16">Finally, the set of <span class="margin_term"><a class="glossterm">real numbers</a><span class="glossdef">The set of all rational and irrational numbers.</span></span>, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0015" display="inline"><mi>ℝ</mi></math></span>, is defined as the set of all rational numbers combined with the set of all irrational numbers. Therefore, all the numbers defined so far are subsets of the set of real numbers. In summary,</p>
<div class="informalfigure large block">
<img src="section_04/b821062d589955109a7ab44751188291.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p18">The set of <span class="margin_term"><a class="glossterm">even integers</a><span class="glossdef">Integers that are divisible by 2.</span></span> is the set of all integers that are evenly divisible by 2. We can obtain the set of even integers by multiplying each integer by 2.</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p19"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0016" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>{</mo><mrow><mn>…</mn><mo>,</mo><mo>−</mo><mn>6</mn><mo>,</mo><mo>−</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mtext> </mtext><mn>0</mn><mo>,</mo><mtext> </mtext><mn>2</mn><mo>,</mo><mtext> </mtext><mn>4</mn><mo>,</mo><mtext> </mtext><mn>6</mn><mo>,</mo><mn>…</mn></mrow><mo>}</mo></mrow><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>g</mi><mi>e</mi><mi>r</mi><mi>s</mi></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p20">The set of <span class="margin_term"><a class="glossterm">odd integers</a><span class="glossdef">Nonzero integers that are not divisible by 2.</span></span> is the set of all nonzero integers that are not evenly divisible by 2.</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0017" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>{</mo><mrow><mn>…</mn><mo>,</mo><mo>−</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mtext> </mtext><mn>1</mn><mo>,</mo><mtext> </mtext><mn>3</mn><mo>,</mo><mtext> </mtext><mn>5</mn><mo>,</mo><mn>…</mn></mrow><mo>}</mo></mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>O</mi><mi>d</mi><mi>d</mi><mtext> </mtext><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>g</mi><mi>e</mi><mi>r</mi><mi>s</mi></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p22">A <span class="margin_term"><a class="glossterm">prime number</a><span class="glossdef">Integer greater than 1 that is divisible only by 1 and itself.</span></span> is an integer greater than 1 that is divisible only by 1 and itself. The smallest prime number is 2 and the rest are necessarily odd.</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0018" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>{</mo><mrow><mn>2</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>3</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>5</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>7</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>11</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>13</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>17</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>19</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>23</mn><mo>,</mo><mn>…</mn></mrow><mo>}</mo></mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi><mi>e</mi><mtext> </mtext><mi>N</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>s</mi></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p24">Any integer greater than 1 that is not prime is called a <span class="margin_term"><a class="glossterm">composite number</a><span class="glossdef">Integers greater than 1 that are not prime.</span></span> and can be uniquely written as a product of primes. When a composite number, such as 42, is written as a product, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0019" display="inline"><mrow><mn>42</mn><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>21</mn></mrow></math></span>, we say that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0020" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>21</mn></mrow></math></span> is a <span class="margin_term"><a class="glossterm">factorization</a><span class="glossdef">Any combination of factors, multiplied together, resulting in the product.</span></span> of 42 and that 2 and 21 are <span class="margin_term"><a class="glossterm">factors</a><span class="glossdef">Any of the numbers that form a product.</span></span>. Note that factors divide the number evenly. We can continue to write composite factors as products until only a product of primes remains.</p>
<div class="informalfigure large block">
<img src="section_04/7fa3d7ca582a44299cacfa004c22a023.png">
</div>
<p class="para block" id="fwk-redden-ch01_s01_s01_p26">Therefore, the <span class="margin_term"><a class="glossterm">prime factorization</a><span class="glossdef">The unique factorization of a natural number written as a product of primes.</span></span> of 42 is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0021" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>7</mn></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch01_s01_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch01_s01_s01_p27">Determine the prime factorization of 210.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s01_p28">Begin by writing 210 as a product with 10 as a factor. Then continue factoring until only a product of primes remains.</p>
<p class="para" id="fwk-redden-ch01_s01_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0022" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>210</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn><mo>⋅</mo><mn>21</mn></mtd></mtr><mtr><mtd columnalign="left"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mo>⋅</mo><mn>5</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="left"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn><mo>⋅</mo><mn>7</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s01_s01_p30">Since the prime factorization is unique, it does not matter how we choose to initially factor the number; the end result will be the same.</p>
<p class="para" id="fwk-redden-ch01_s01_s01_p31">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0023" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn><mo>⋅</mo><mn>7</mn></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch01_s01_s01_p32">A <span class="margin_term"><a class="glossterm">fraction</a><span class="glossdef">A rational number written as a quotient of two integers: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0024" display="inline"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0025" display="inline"><mrow><mi>b</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span> is a rational number written as a quotient, or ratio, of two integers <em class="emphasis">a</em> and <em class="emphasis">b</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0026" display="inline"><mrow><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></math></span>.</p>
<div class="informalfigure large block">
<img src="section_04/479b213137bd9f76ec1d37c83d23066a.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p34">The integer above the fraction bar is called the <span class="margin_term"><a class="glossterm">numerator</a><span class="glossdef">The number above the fraction bar.</span></span> and the integer below is called the <span class="margin_term"><a class="glossterm">denominator</a><span class="glossdef">The number below the fraction bar.</span></span>. Two equal ratios expressed using different numerators and denominators are called <span class="margin_term"><a class="glossterm">equivalent fractions</a><span class="glossdef">Two equal fractions expressed using different numerators and denominators.</span></span>. For example,</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p35"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0027" display="block"><mrow><mfrac><mrow><mn>50</mn></mrow><mrow><mn>100</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p36">Consider the following factorizations of 50 and 100:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p37"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0028" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>50</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mo>⋅</mo><mn>25</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>100</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mo>⋅</mo><mn>25</mn></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p38">The numbers 50 and 100 share the factor 25. A shared factor is called a <span class="margin_term"><a class="glossterm">common factor</a><span class="glossdef">A factor that is shared by more than one real number.</span></span>. Making use of the fact that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0029" display="inline"><mrow><mfrac><mrow><mn>25</mn></mrow><mrow><mn>25</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>, we have</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p39"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0030" display="block"><mrow><mfrac><mrow><mn>50</mn></mrow><mrow><mn>100</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo>⋅</mo><menclose notation="updiagonalstrike"><mrow><mn>25</mn></mrow></menclose></mrow><mrow><mn>4</mn><mo>⋅</mo><menclose notation="updiagonalstrike"><mrow><mn>25</mn></mrow></menclose></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>4</mn></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>=</mo><mfrac><mn>2</mn><mn>4</mn></mfrac></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p40">Dividing <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0031" display="inline"><mrow><mfrac><mrow><mn>25</mn></mrow><mrow><mn>25</mn></mrow></mfrac></mrow></math></span> and replacing this factor with a 1 is called <span class="margin_term"><a class="glossterm">cancelling</a><span class="glossdef">The process of dividing out common factors in the numerator and the denominator.</span></span>. Together, these basic steps for finding equivalent fractions define the process of <span class="margin_term"><a class="glossterm">reducing</a><span class="glossdef">The process of finding equivalent fractions by dividing the numerator and the denominator by common factors.</span></span>. Since factors divide their product evenly, we achieve the same result by dividing both the numerator and denominator by 25 as follows:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p41"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0032" display="block"><mrow><mfrac><mrow><mi> </mi><mi> </mi><mn>50</mn><mstyle color="#007fbf"><mo>÷</mo><mn>25</mn></mstyle></mrow><mrow><mn>100</mn><mstyle color="#007fbf"><mo>÷</mo><mn>25</mn></mstyle></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>4</mn></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p42">Finding equivalent fractions where the numerator and denominator are <span class="margin_term"><a class="glossterm">relatively prime</a><span class="glossdef">Numbers that have no common factor other than 1.</span></span>, or have no common factor other than 1, is called <span class="margin_term"><a class="glossterm">reducing to lowest terms</a><span class="glossdef">Finding equivalent fractions where the numerator and the denominator share no common integer factor other than 1.</span></span>. This can be done by dividing the numerator and denominator by the <span class="margin_term"><a class="glossterm">greatest common factor (GCF).</a><span class="glossdef">The largest shared factor of any number of integers.</span></span> The GCF is the largest number that divides a set of numbers evenly. One way to find the GCF of 50 and 100 is to list all the factors of each and identify the largest number that appears in both lists. Remember, each number is also a factor of itself.</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p43"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0033" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>{</mo><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>5</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>10</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>25</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>50</mn></mstyle></mrow><mo>}</mo></mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>F</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mtext> </mtext><mn>50</mn></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>{</mo><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo>,</mo><mn>4</mn><mo>,</mo><mstyle mathvariant="bold"><mn>5</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>10</mn></mstyle><mo>,</mo><mn>20</mn><mo>,</mo><mstyle mathvariant="bold"><mn>25</mn></mstyle><mo>,</mo><mstyle mathvariant="bold"><mn>50</mn></mstyle><mo>,</mo><mn>100</mn></mrow><mo>}</mo></mrow><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>F</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mtext> </mtext><mn>100</mn></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p44">Common factors are listed in bold, and we see that the greatest common factor is 50. We use the following notation to indicate the GCF of two numbers: GCF(50, 100) = 50. After determining the GCF, reduce by dividing both the numerator and the denominator as follows:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p45"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0034" display="block"><mrow><mfrac><mrow><mi> </mi><mi> </mi><mn>50</mn><mstyle color="#007fbf"><mo>÷</mo><mn>50</mn></mstyle></mrow><mrow><mn>100</mn><mstyle color="#007fbf"><mo>÷</mo><mn>50</mn></mstyle></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch01_s01_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch01_s01_s01_p46">Reduce to lowest terms: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0035" display="inline"><mrow><mfrac><mrow><mn>108</mn></mrow><mrow><mn>72</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s01_p47">A quick way to find the GCF of the numerator and denominator requires us to first write each as a product of primes. The GCF will be the product of all the common prime factors.</p>
<p class="para" id="fwk-redden-ch01_s01_s01_p48"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0036" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd><mn>108</mn><mo>=</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>⋅</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>72</mn><mo>=</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>⋅</mo><mn>2</mn><mo>⋅</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>3</mn></mstyle></mtd></mtr></mtable><mo>}</mo></mrow><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>GCF(108, 72) = </mtext><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>⋅</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>=</mo><mn>36</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s01_s01_p49">In this case, the product of the common prime factors is 36.</p>
<p class="para" id="fwk-redden-ch01_s01_s01_p50"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0037" display="block"><mrow><mfrac><mrow><mn>108</mn></mrow><mrow><mn>72</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>108</mn><mstyle color="#007fbf"><mo>÷</mo><mn>36</mn></mstyle></mrow><mrow><mn>72</mn><mstyle color="#007fbf"><mo>÷</mo><mn>36</mn></mstyle></mrow></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s01_s01_p51">We can convert the improper fraction <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0038" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span> to a mixed number <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0039" display="inline"><mrow><mn>1</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>; however, it is important to note that converting to a mixed number is not part of the reducing process. We consider improper fractions, such as <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0040" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>, to be reduced to lowest terms. In algebra it is often preferable to work with improper fractions, although in some applications, mixed numbers are more appropriate.</p>
<p class="para" id="fwk-redden-ch01_s01_s01_p52">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0041" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p53">Recall the relationship between multiplication and division:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p54"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0042" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mstyle color="#007fbf"><mi>d</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mo>→</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd></mtr><mtr><mtd><mrow><mstyle color="#007fbf"><mi>d</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mo>→</mo></mstyle></mrow></mtd></mtr></mtable><mfrac><mrow><mn>12</mn></mrow><mn>6</mn></mfrac><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext> </mtext><mo>←</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mstyle><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>because</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mn>6</mn><mo>⋅</mo><mn>2</mn><mo>=</mo><mn>12</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p55">In this case, the <span class="margin_term"><a class="glossterm">dividend</a><span class="glossdef">A number to be divided by another number.</span></span> 12 is evenly divided by the <span class="margin_term"><a class="glossterm">divisor</a><span class="glossdef">The number that is divided into the dividend.</span></span> 6 to obtain the <span class="margin_term"><a class="glossterm">quotient</a><span class="glossdef">The result of division.</span></span> 2. It is true in general that if we multiply the divisor by the quotient we obtain the dividend. Now consider the case where the dividend is zero and the divisor is nonzero:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p56"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0043" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mfrac><mn>0</mn><mn>6</mn></mfrac><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>since</mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mn>6</mn><mo>⋅</mo><mn>0</mn><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p57">This demonstrates that zero divided by any nonzero real number must be zero. Now consider a nonzero number divided by zero:</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p58"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0044" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mfrac><mrow><mn>12</mn></mrow><mn>0</mn></mfrac><mo>=</mo><mi> </mi><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mn>0</mn><mo>⋅</mo><mi> </mi><mstyle color="#007fbf"><mo>?</mo></mstyle><mo>=</mo><mn>12</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p59">Zero times anything is zero and we conclude that there is no real number such that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0045" display="inline"><mrow><mn>0</mn><mo>⋅</mo><mi> </mi><mo>?</mo><mo>=</mo><mn>12</mn></mrow><mo>.</mo></math></span> Thus, the quotient <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0046" display="inline"><mrow><mn>12</mn><mo>÷</mo><mn>0</mn></mrow></math></span> is <span class="margin_term"><a class="glossterm">undefined</a><span class="glossdef">A quotient such as <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0047" display="inline"><mrow><mfrac><mn>5</mn><mn>0</mn></mfrac></mrow></math></span> is left without meaning and is not assigned an interpretation.</span></span>. Try it on a calculator, what does it say? For our purposes, we will simply write “undefined.” To summarize, given any real number <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0048" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, then</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p60"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0049" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mn>0</mn><mtext> </mtext><mtext> </mtext><mo>÷</mo><mi>a</mi><mo>=</mo><mfrac><mn>0</mn><mi>a</mi></mfrac><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mi>z</mi><mi>e</mi><mi>r</mi><mi>o</mi></mstyle><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>a</mi><mo>÷</mo><mtext> </mtext><mn>0</mn><mo>=</mo><mfrac><mi>a</mi><mn>0</mn></mfrac><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mi>u</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>f</mi><mi>i</mi><mi>n</mi><mi>e</mi><mi>d</mi></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s01_p61">We are left to consider the case where the dividend and divisor are both zero.</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p62"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0050" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mfrac><mn>0</mn><mn>0</mn></mfrac><mo>=</mo><mi> </mi><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mn>0</mn><mo>⋅</mo><mi> </mi><mstyle color="#007fbf"><mo>?</mo></mstyle><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p63">Here, any real number seems to work. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0051" display="inline"><mrow><mn>0</mn><mo>⋅</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span> and also, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0052" display="inline"><mrow><mn>0</mn><mo>⋅</mo><mi> </mi><mn>3</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span> Therefore, the quotient is uncertain or <span class="margin_term"><a class="glossterm">indeterminate</a><span class="glossdef">A quotient such as <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0053" display="inline"><mrow><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow></math></span> is a quantity that is uncertain or ambiguous.</span></span>.</p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p64"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0054" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mn>0</mn><mo>÷</mo><mn>0</mn><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac><mi> </mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mstyle color="#007fbf"><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s01_p65">In this course, we state that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0055" display="inline"><mrow><mn>0</mn><mo>÷</mo><mn>0</mn></mrow></math></span> is undefined.</p>
</div>
<div class="section" id="fwk-redden-ch01_s01_s02" version="5.0" lang="en">
<h2 class="title editable block">The Number Line and Notation</h2>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p01">A <span class="margin_term"><a class="glossterm">real number line</a><span class="glossdef">A line that allows us to visually represent real numbers by associating them with points on the line.</span></span>, or simply <strong class="emphasis bold">number line</strong>, allows us to visually display real numbers by associating them with unique points on a line. The real number associated with a point is called a <span class="margin_term"><a class="glossterm">coordinate</a><span class="glossdef">The real number associated with a point on a number line.</span></span>. A point on the real number line that is associated with a coordinate is called its <span class="margin_term"><a class="glossterm">graph</a><span class="glossdef">A point on the number line associated with a coordinate.</span></span>. To construct a number line, draw a horizontal line with arrows on both ends to indicate that it continues without bound. Next, choose any point to represent the number zero; this point is called the <span class="margin_term"><a class="glossterm">origin</a><span class="glossdef">The point on the number line that represents zero.</span></span>.</p>
<div class="informalfigure large block">
<img src="section_04/444d79554b28a14fec2e9340414807ac.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p03">Positive real numbers lie to the right of the origin and negative real numbers lie to the left. The number zero (0) is neither positive nor negative. Typically, each tick represents one unit.</p>
<div class="informalfigure large block">
<img src="section_04/b360022849f5b10990aaf67392f47bb5.png">
</div>
<p class="para block" id="fwk-redden-ch01_s01_s02_p05">As illustrated below, the scale need not always be one unit. In the first number line, each tick mark represents two units. In the second, each tick mark represents <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0056" display="inline"><mrow><mfrac><mn>1</mn><mn>7</mn></mfrac></mrow></math></span>:</p>
<div class="informalfigure large block">
<img src="section_04/c6cd5370b59b03018335a3241d932712.png">
</div>
<p class="para block" id="fwk-redden-ch01_s01_s02_p07">The graph of each real number is shown as a dot at the appropriate point on the number line. A partial graph of the set of integers <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0057" display="inline"><mrow><mi> </mi><mi>ℤ</mi></mrow></math></span>, follows:</p>
<div class="informalfigure large block">
<img src="section_04/6d4b0db4f2100837b1d03567b974bf3e.png">
</div>
<div class="callout block" id="fwk-redden-ch01_s01_s02_n01">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch01_s01_s02_p09">Graph the following set of real numbers: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0058" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><mi> </mi><mi> </mi><mn>0</mn><mo>,</mo><mi> </mi><mi> </mi><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mn>2</mn></mrow><mo>}</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p10">Graph the numbers on a number line with a scale where each tick mark represents <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0059" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> unit.</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p11">Answer: </p>
<div class="informalfigure large">
<img src="section_04/f4975994065f242af522194c337f6f82.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p12">The <span class="margin_term"><a class="glossterm">opposite</a><span class="glossdef">Real numbers whose graphs are on opposite sides of the origin with the same distance to the origin.</span></span> of any real number <em class="emphasis">a</em> is −<em class="emphasis">a</em>. Opposite real numbers are the same distance from the origin on a number line, but their graphs lie on opposite sides of the origin and the numbers have opposite signs.</p>
<div class="informalfigure large block">
<img src="section_04/f77faab3734eea6566dbf0dd1e6e4324.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p14">Given the integer −7, the integer the same distance from the origin and with the opposite sign is +7, or just 7.</p>
<div class="informalfigure large block">
<img src="section_04/219b42c19b2c2e9e5b76d529a8cd3f4c.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p16">Therefore, we say that the opposite of −7 is −(−7) = 7. This idea leads to what is often referred to as the <span class="margin_term"><a class="glossterm">double-negative property</a><span class="glossdef">The opposite of a negative number is positive: −(−<em class="emphasis">a</em>) = <em class="emphasis">a</em>.</span></span>. For any real number <em class="emphasis">a</em>,</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p17"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0060" display="block"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mi>a</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>a</mi></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch01_s01_s02_n02">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch01_s01_s02_p18">Calculate: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0061" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p19">Here we apply the double-negative within the innermost parentheses first.</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p20"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0062" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mn>8</mn></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s01_s02_p21">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0063" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p22">In general, an odd number of sequential negative signs results in a negative value and an even number of sequential negative signs results in a positive value.</p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p23">When comparing real numbers on a number line, the larger number will always lie to the right of the smaller one. It is clear that 15 is greater than 5, but it may not be so clear to see that −1 is greater than −5 until we graph each number on a number line.</p>
<div class="informalfigure large block">
<img src="section_04/0e7f53bc2b33c163e3e6f2184bc6d1b5.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p25">We use symbols to help us efficiently communicate relationships between numbers on the number line.</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p26"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0064" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>t</mi><mi>y</mi><mtext> </mtext><mi>R</mi><mi>e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>h</mi><mi>i</mi><mi>p</mi><mi>s</mi></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mstyle color="#007fbf"><mi>O</mi><mi>r</mi><mi>d</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>R</mi><mi>e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>h</mi><mi>i</mi><mi>p</mi><mi>s</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>=</mo><mtext> "</mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext>"</mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mo><</mo><mtext> "</mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>s</mi><mi>s</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext>"</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>≠</mo><mtext> "</mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext>"</mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mo>></mo><mtext> "</mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>g</mi><mi>r</mi><mi>e</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext>"</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>≈</mo><mtext> "</mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>a</mi><mi>p</mi><mi>p</mi><mi>r</mi><mi>o</mi><mi>x</mi><mi>i</mi><mi>m</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext>"</mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mo>≤</mo><mtext> "</mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>s</mi><mi>s</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext>"</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mo>≥</mo><mtext> "</mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>g</mi><mi>r</mi><mi>e</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext>"</mtext></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p27">The relationship between the <span class="margin_term"><a class="glossterm">integers</a><span class="glossdef">The set of positive and negative whole numbers combined with zero: {…, −3, −2, −1, 0, 1, 2, 3, …}.</span></span> in the previous illustration can be expressed two ways as follows:</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p28"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0065" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mo>−</mo><mn>5</mn><mo><</mo><mo>−</mo><mn>1</mn><mtext> </mtext></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mtext> </mtext><mtext>"</mtext><mi>N</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>s</mi><mi>s</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext> </mtext><mi>n</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>o</mi><mi>n</mi><mi>e</mi><mo>.</mo><mtext>"</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mn>1</mn><mo>></mo><mo>−</mo><mn>5</mn><mtext> </mtext></mrow></mtd><mtd><mstyle color="#007fbf"><mrow><mtext> </mtext><mtext>"</mtext><mi>N</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>o</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>g</mi><mi>r</mi><mi>e</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext> </mtext><mi>n</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>i</mi><mi>v</mi><mi>e</mi><mo>.</mo><mtext>"</mtext></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p29">The symbols < and > are used to denote <span class="margin_term"><a class="glossterm">strict inequalities</a><span class="glossdef">Express ordering relationships using the symbol < for “less than” and > for “greater than.”</span></span>, and the symbols <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0066" display="inline"><mo>≤</mo></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0067" display="inline"><mo>≥</mo></math></span> are used to denote <span class="margin_term"><a class="glossterm">inclusive inequalities</a><span class="glossdef">Use the symbol <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0068" display="inline"><mo>≤</mo></math></span> to express quantities that are “less than or equal to” and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0069" display="inline"><mo>≥</mo></math></span> for quantities that are “greater than or equal to” each other.</span></span>. In some situations, more than one symbol can be correctly applied. For example, the following two statements are both true:</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p30"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0070" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mo>−</mo><mn>10</mn><mo><</mo><mn>0</mn></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mo>−</mo><mn>10</mn><mo>≤</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p31">In addition, the “<em class="emphasis">or equal to</em>” component of an inclusive inequality allows us to correctly write the following:</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p32"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0071" display="block"><mrow><mo>−</mo><mn>10</mn><mo>≤</mo><mo>−</mo><mn>10</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p33">The logical use of the word “<em class="emphasis">or</em>” requires that only one of the conditions need be true: the “<em class="emphasis">less than</em>” or the “<em class="emphasis">equal to</em>.”</p>
<div class="callout block" id="fwk-redden-ch01_s01_s02_n03">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch01_s01_s02_p34">Fill in the blank with <, =, or >: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0072" display="inline"><mrow><mo>−</mo><mn>2</mn><mtext>_</mtext><mtext>_</mtext><mtext>_</mtext><mo>−</mo><mn>12</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p35">Use > because the graph of −2 is to the right of the graph of −12 on a number line. Therefore, −2 > −12, which reads, “<em class="emphasis">negative two is greater than negative twelve.</em>”</p>
<div class="informalfigure large">
<img src="section_04/634b7c3cb166ca9c388a07217f0c88b4.png">
</div>
<p class="para" id="fwk-redden-ch01_s01_s02_p37">Answer: −2 > −12</p>
</div>
<p class="para block" id="fwk-redden-ch01_s01_s02_p38">An <span class="margin_term"><a class="glossterm">algebraic inequality</a><span class="glossdef">Algebraic expressions related with the symbols <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0073" display="inline"><mo>≤</mo></math></span>, <, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0074" display="inline"><mo>≥</mo></math></span>, and >.</span></span>, such as <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0075" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, is read, “<em class="emphasis">x is greater than or equal to 2</em>.” Here the letter <em class="emphasis">x</em> is a variable, which can represent any real number. However, the statement <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0076" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>2</mn></mrow></math></span> imposes a condition on the variable. <span class="margin_term"><a class="glossterm">Solutions</a><span class="glossdef">Values that can be used in place of the variable to satisfy the given condition.</span></span> are the values for <em class="emphasis">x</em> that satisfy the condition. This inequality has infinitely many solutions for <em class="emphasis">x</em>, some of which are 2, 3, 4.1, 5, 20, and 20.001. Since it is impossible to list all of the solutions, a system is needed that allows a clear communication of this infinite set. Common ways of expressing solutions to an inequality are by graphing them on a number line, using interval notation, or using set notation.</p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p39">To express the solution graphically, draw a number line and shade in all the values that are solutions to the inequality. This is called the <span class="margin_term"><a class="glossterm">graph of the solution set</a><span class="glossdef">Solutions to an algebraic expression expressed on a number line.</span></span>. Interval and set notation follow:</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p40"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0077" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext>"</mtext><mi>x</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>g</mi><mi>r</mi><mi>e</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mn>2</mn><mtext>"</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_04/59f55fe30d750b3f5d1ab0be9b956339.png">
</div>
<p class="para block" id="fwk-redden-ch01_s01_s02_p42"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0078" display="block"><mrow><mtable columnspacing="0.1em" columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>:</mo></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mrow><mo>[</mo><mrow><mn>2</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>e</mi><mi>t</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>:</mo></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mrow><mo>{</mo><mrow><mi>x</mi><mo>∈</mo><mi>ℝ</mi><mo>|</mo><mi>x</mi><mo>≥</mo><mn>2</mn></mrow><mo>}</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p43">In this example, there is an inclusive inequality, which means that the lower-bound 2 is included in the solution set. Denote this with a closed dot on the number line and a square bracket in interval notation. The symbol ∞ is read as “<span class="margin_term"><a class="glossterm">infinity</a><span class="glossdef">The symbol ∞ indicates the interval is unbounded to the right.</span></span>” and indicates that the set is unbounded to the right on a number line. If using a standard keyboard, use (inf) as a shortened form to denote infinity. Now compare the notation in the previous example to that of the strict, or noninclusive, inequality that follows:</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p44"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0079" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mo>"</mo><mi>x</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>s</mi><mi>s</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>a</mi><mi>n</mi><mtext> </mtext><mn>3</mn><mo>"</mo><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo><</mo><mn>3</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_04/43ffcd6ab107cb876730f16028f77b94.png">
</div>
<p class="para block" id="fwk-redden-ch01_s01_s02_p46"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0080" display="block"><mrow><mtable columnspacing="0.1em" columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>:</mo></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>e</mi><mi>t</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>:</mo></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mrow><mo>{</mo><mrow><mi>x</mi><mo>∈</mo><mi>ℝ</mi><mo>|</mo><mi>x</mi><mo><</mo><mn>3</mn></mrow><mo>}</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p47">Strict inequalities imply that solutions may get very close to the boundary point, in this case 3, but not actually include it. Denote this idea with an open dot on the number line and a round parenthesis in interval notation. The symbol −∞ is read as “<span class="margin_term"><a class="glossterm">negative infinity</a><span class="glossdef">The symbol −∞ indicates the interval is unbounded to the left.</span></span>” and indicates that the set is unbounded to the left on a number line. Infinity is a bound to the real numbers, but is not itself a real number: it cannot be included in the solution set and thus is always enclosed with a parenthesis.</p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p48">Interval notation is textual and is determined after graphing the solution set on a number line. The numbers in interval notation should be written in the same order as they appear on the number line, with smaller numbers in the set appearing first. Set notation, sometimes called set-builder notation, allows us to describe the set using familiar mathematical notation. For example,</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p49"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0081" display="block"><mrow><mrow><mo>{</mo><mrow><mi>x</mi><mo>∈</mo><mi>ℝ</mi><mo>|</mo><mi>x</mi><mo>≥</mo><mn>2</mn></mrow><mo>}</mo></mrow></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p50">Here, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0082" display="inline"><mrow><mi>x</mi><mo>∈</mo><mi>ℝ</mi></mrow></math></span> describes the type of number. This implies that the variable <em class="emphasis">x</em> represents a real number. The statement <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0083" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>2</mn></mrow></math></span> is the condition that describes the set using mathematical notation. At this point in our study of algebra, it is assumed that all variables represent real numbers. For this reason, you can omit the “<span class="inlineequation"><math xml:id="fwk-redden-ch01_m0084" display="inline"><mrow><mo>∈</mo><mi>ℝ</mi></mrow></math></span>”, and write</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p51"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0085" display="block"><mrow><mrow><mo>{</mo><mrow><mi>x</mi><mo>|</mo><mi>x</mi><mo>≥</mo><mn>2</mn></mrow><mo>}</mo></mrow></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch01_s01_s02_n04">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch01_s01_s02_p52">Graph the solution set and give the interval and set notation equivalents: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0086" display="inline"><mrow><mi>x</mi><mo><</mo><mo>−</mo><mn>20</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p53">Use an open dot at −20, because of the strict inequality <, and shade all real numbers to the left.</p>
<div class="informalfigure large">
<img src="section_04/074b84275d25a592b102cce012569afe.png">
</div>
<p class="para" id="fwk-redden-ch01_s01_s02_p55">Answer: Interval notation: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0087" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>20</mn></mrow><mo>)</mo></mrow></mrow></math></span>; set notation: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0088" display="inline"><mrow><mrow><mo>{</mo><mrow><mi>x</mi><mo>|</mo><mi>x</mi><mo><</mo><mo>−</mo><mn>20</mn></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch01_s01_s02_p56">A <span class="margin_term"><a class="glossterm">compound inequality</a><span class="glossdef">Two or more inequalities in one statement joined by the word “and” or by the word “or.”</span></span> is actually two or more inequalities in one statement joined by the word “and” or by the word “or”. Compound inequalities with the logical “or” require that either condition must be satisfied. Therefore, the solution set of this type of compound inequality consists of all the elements of the solution sets of each inequality. When we join these individual solution sets it is called the <span class="margin_term"><a class="glossterm">union</a><span class="glossdef">The set formed by joining the individual solution sets indicated by the logical use of the word “or” and denoted with the symbol <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0089" display="inline"><mo>∪</mo><mo>.</mo></math></span></span></span>, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0090" display="inline"><mrow><mi> </mi><mo>∪</mo></mrow><mo>.</mo></math></span> For example,</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p57"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0091" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo><</mo><mn>3</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mi>x</mi><mo>≥</mo><mn>6</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_04/6f69a71d6784abbd56e0939bede5fccc.png">
</div>
<p class="para block" id="fwk-redden-ch01_s01_s02_p59"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0092" display="block"><mrow><mtable columnspacing="0.1em" columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>:</mo></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>[</mo><mrow><mn>6</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>e</mi><mi>t</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>:</mo></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mrow><mo>{</mo><mrow><mi>x</mi><mo>|</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mo><</mo><mn>3</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>or</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>6</mn></mrow><mo>}</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p60">An inequality such as,
<span class="informalequation"><math xml:id="fwk-redden-ch01_m0093" display="block"><mrow><mo>−</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo><</mo><mn>3</mn></mrow></math></span>
reads, “<em class="emphasis">negative one is less than or equal to x and x is less than three</em>.” This is actually a compound inequality because it can be decomposed as follows:</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p63"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0094" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mo>−</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo><</mo><mn>3</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p64">The logical “and” requires that both conditions must be true. Both inequalities will be satisfied by all the elements in the <span class="margin_term"><a class="glossterm">intersection</a><span class="glossdef">The set formed by the shared values of the individual solution sets that is indicated by the logical use of the word “and,” denoted with the symbol <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0095" display="inline"><mo>∩</mo><mo>.</mo></math></span></span></span>, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0096" display="inline"><mrow><mi> </mi><mo>∩</mo></mrow></math></span>, of the solution sets of each.</p>
<div class="callout block" id="fwk-redden-ch01_s01_s02_n05">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch01_s01_s02_p65">Graph and give the interval notation equivalent: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0097" display="inline"><mrow><mo>−</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo><</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p66">Determine the intersection, or overlap, of the two solution sets to <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0098" display="inline"><mrow><mi>x</mi><mo><</mo><mn>3</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0099" display="inline"><mrow><mi>x</mi><mo>≥</mo><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span> The solutions to each inequality are sketched above the number line as a means to determine the intersection, which is graphed on the number line below.</p>
<div class="informalfigure large">
<img src="section_04/7f0dfa74ecceb3e0f6a8bc46785099ef.png">
</div>
<p class="para" id="fwk-redden-ch01_s01_s02_p68">Here, 3 is not a solution because it solves only one of the inequalities. Alternatively, we may interpret <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0100" display="inline"><mrow><mo>−</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo><</mo><mn>3</mn></mrow></math></span> as all possible values for <em class="emphasis">x</em> between, or bounded by, −1 and 3 where −1 is included in the solution set.</p>
<p class="para" id="fwk-redden-ch01_s01_s02_p69">Answer: Interval notation: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0101" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mi> </mi><mtext> </mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span>; set notation: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0102" display="inline"><mrow><mrow><mo>{</mo><mrow><mi>x</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo><</mo><mn>3</mn></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p70">In this text, we will often point out the equivalent notation used to express mathematical quantities electronically using the standard symbols available on a keyboard.</p>
<p class="para block" id="fwk-redden-ch01_s01_s02_p71"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0103" display="block"><mrow><mtable columnspacing="0.1em" columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><mo>×</mo><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mtext>"</mtext><mtext> </mtext><mtext>*</mtext><mtext> </mtext><mtext>"</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mo>≥</mo><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mtext>" >= "</mtext></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mo>÷</mo><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mtext>"</mtext><mtext> </mtext><mtext>/</mtext><mtext> </mtext><mtext>"</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mo>≤</mo><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mtext>" <= "</mtext></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>≠</mo><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mtext>" </mtext><mtext>!</mtext><mtext>= "</mtext></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s02_p72">Many calculators, computer algebra systems, and programming languages use the notation presented above, in quotes.</p>
</div>
<div class="section" id="fwk-redden-ch01_s01_s03" version="5.0" lang="en">
<h2 class="title editable block">Absolute Value</h2>
<p class="para block" id="fwk-redden-ch01_s01_s03_p01">The <span class="margin_term"><a class="glossterm">absolute value</a><span class="glossdef">The absolute value of a number represents the distance from the graph of the number to zero on a number line.</span></span> of a real number <em class="emphasis">a</em>, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0104" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow></mrow></math></span>, is defined as the distance between zero (the origin) and the graph of that real number on the number line. Since it is a distance, it is always positive. For example,</p>
<p class="para block" id="fwk-redden-ch01_s01_s03_p02"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0105" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mrow><mo>|</mo><mn>4</mn><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s03_p03">Both 4 and −4 are four units from the origin, as illustrated below:</p>
<div class="informalfigure large block">
<img src="section_04/b66f32c5f679d17c46abf74b78016fa4.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s01_s03_p05">Also, it is worth noting that,</p>
<p class="para block" id="fwk-redden-ch01_s01_s03_p06"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0106" display="block"><mrow><mrow><mo>|</mo><mn>0</mn><mo>|</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s03_p07">The algebraic definition of the absolute value of a real number <em class="emphasis">a</em> follows:</p>
<p class="para block" id="fwk-redden-ch01_s01_s03_p08"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0107" display="block"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>a</mi><mi> </mi><mi> </mi><mi> </mi><mtext>if</mtext><mi> </mi><mi> </mi><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mi>a</mi><mi> </mi><mi> </mi><mi> </mi><mtext>if</mtext><mi> </mi><mi> </mi><mi> </mi><mi>a</mi><mo><</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s03_p09">This is called a <span class="margin_term"><a class="glossterm">piecewise definition</a><span class="glossdef">A definition that changes depending on the value of the variable.</span></span>. The result depends on the quantity <em class="emphasis">a</em>. If <em class="emphasis">a</em> is nonnegative, as indicated by the inequality <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0108" display="inline"><mrow><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, then the absolute value will be that number <em class="emphasis">a</em>. If <em class="emphasis">a</em> is negative, as indicated by the inequality <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0109" display="inline"><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></math></span>, then the absolute value will be the opposite of that number, −<em class="emphasis">a</em>. The results will be the same as the geometric definition. For example, to determine <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0110" display="inline"><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mrow></math></span> we make note that the value is negative and use the second part of the definition. The absolute value will be the opposite of −4.</p>
<p class="para block" id="fwk-redden-ch01_s01_s03_p10"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0111" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s01_s03_p11">At this point, we can determine what real numbers have certain absolute values.</p>
<div class="callout block" id="fwk-redden-ch01_s01_s03_n01">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch01_s01_s03_p12">Determine the values represented by <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0112" display="inline"><mrow><mrow><mo>|</mo><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow><mo>|</mo></mrow><mo>=</mo><mn>6</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s03_p13">Think of a real number whose distance to the origin is 6 units. There are two solutions: the distance to the right of the origin and the distance to the left of the origin, namely <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0113" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>±</mo><mn>6</mn></mrow><mo>}</mo></mrow></mrow><mo>.</mo></math></span> The symbol ± is read “<em class="emphasis">plus or minus</em>” and indicates that there are two answers, one positive and one negative.</p>
<p class="para" id="fwk-redden-ch01_s01_s03_p14"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0114" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>6</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>6</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mrow><mo>|</mo><mn>6</mn><mo>|</mo></mrow><mo>=</mo><mn>6</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s01_s03_p15">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0115" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>±</mo><mn>6</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s01_s03_n02">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch01_s01_s03_p16">Determine the values represented by <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0116" display="inline"><mrow><mrow><mo>|</mo><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow><mo>|</mo></mrow><mo>=</mo><mo>−</mo><mn>6</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s03_p17">Here we wish to find a value where the distance to the origin is negative. Since negative distance is not defined, this equation has no solution. Use the empty set Ø to denote this.</p>
<p class="para" id="fwk-redden-ch01_s01_s03_p18">Answer: Ø</p>
</div>
<p class="para block" id="fwk-redden-ch01_s01_s03_p19">The absolute value can be expressed textually using the notation abs(<em class="emphasis">a</em>). We often encounter negative absolute values, such as <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0117" display="inline"><mrow><mo>−</mo><mrow><mo>|</mo><mn>3</mn><mo>|</mo></mrow></mrow></math></span> or −abs(3). Notice that the negative sign is in front of the absolute value symbol. In this case, work the absolute value first and then find the opposite of the result.</p>
<p class="para block" id="fwk-redden-ch01_s01_s03_p20"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0118" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd></mtd><mtd columnalign="left"><mo>−</mo><mrow><mo>|</mo><mn>3</mn><mo>|</mo></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd columnalign="center"><mo> </mo><mo> </mo><mtext>and</mtext><mo> </mo><mo> </mo></mtd><mtd></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch01_s01_s03_p21">Try not to confuse this with the double negative property, which states that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0119" display="inline"><mrow><mo>−</mo><mi> </mi><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mi> </mi><mo>=</mo><mi> </mi><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch01_s01_s03_n03">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch01_s01_s03_p22">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0120" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mtext>−</mtext><mrow><mo>|</mo><mrow><mo>−</mo><mn>50</mn></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s01_s03_p23">First, find the absolute value of −50 and then apply the double-negative property.</p>
<p class="para" id="fwk-redden-ch01_s01_s03_p24"><span class="informalequation"><math xml:id="fwk-redden-ch01_m0121" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mrow><mo>(</mo><mrow><mtext>−</mtext><mrow><mstyle color="#007fbf"><mo>|</mo><mrow><mo>−</mo><mn>50</mn></mrow><mo>|</mo></mstyle></mrow></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>50</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>50</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s01_s03_p25">Answer: 50</p>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch01_s01_s03_n04">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch01_s01_s03_l01" mark="bullet">
<li>Algebra is often described as the generalization of arithmetic. The systematic use of variables, used to represent real numbers, allows us to communicate and solve a wide variety of real-world problems. Therefore, it is important to review the subsets of real numbers and their properties.</li>
<li>The number line allows us to visually display real numbers by associating them with unique points on a line.</li>
<li>Special notation is used to communicate equality and order relationships between numbers on a number line.</li>
<li>The absolute value of a real number is defined geometrically as the distance between zero and the graph of that number on a number line. Alternatively, the absolute value of a real number is defined algebraically in a piecewise manner. If a real number <em class="emphasis">a</em> is nonnegative, then the absolute value will be that number <em class="emphasis">a</em>. If <em class="emphasis">a</em> is negative, then the absolute value will be the opposite of that number, −<em class="emphasis">a</em>.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch01_s01_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd01">
<h3 class="title">Part A: Real Numbers</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch01_s01_qs01_p01"><strong class="emphasis bold">Use set notation to list the described elements.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p02">Every other positive odd number up to 21.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p04">Every other positive even number up to 22.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p06">The even prime numbers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p08">Rational numbers that are also irrational.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p10">The set of negative integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p12">The set of negative even integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p14">Three consecutive odd integers starting with 13.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p16">Three consecutive even integers starting with 22.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd01_qd02" start="9">
<p class="para" id="fwk-redden-ch01_s01_qs01_p18"><strong class="emphasis bold">Determine the prime factorization of the given composite number.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p19">195</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p21">78</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p23">330</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p25">273</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p27">180</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p29">350</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd01_qd03" start="15">
<p class="para" id="fwk-redden-ch01_s01_qs01_p31"><strong class="emphasis bold">Reduce to lowest terms.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0128" display="inline"><mrow><mfrac><mrow><mn>42</mn></mrow><mrow><mn>30</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0130" display="inline"><mrow><mfrac><mrow><mn>105</mn></mrow><mrow><mn>70</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0132" display="inline"><mrow><mfrac><mrow><mn>84</mn></mrow><mrow><mn>120</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0134" display="inline"><mrow><mfrac><mrow><mn>315</mn></mrow><mrow><mn>420</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0136" display="inline"><mrow><mfrac><mrow><mn>60</mn></mrow><mrow><mn>45</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0138" display="inline"><mrow><mfrac><mrow><mn>144</mn></mrow><mrow><mn>120</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0140" display="inline"><mrow><mfrac><mrow><mn>64</mn></mrow><mrow><mn>128</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0142" display="inline"><mrow><mfrac><mrow><mn>72</mn></mrow><mrow><mn>216</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0144" display="inline"><mrow><mfrac><mn>0</mn><mrow><mn>25</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0145" display="inline"><mrow><mfrac><mrow><mn>33</mn></mrow><mn>0</mn></mfrac></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02">
<h3 class="title">Part B: Number Line and Notation</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02_qd01" start="25">
<p class="para" id="fwk-redden-ch01_s01_qs01_p52"><strong class="emphasis bold">Graph the following sets of numbers.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p53">{−5, 5, 10, 15}</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p55">{−4, −2, 0, 2, 4}</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0146" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mn>0</mn><mo>,</mo><mtext> </mtext><mn>1</mn><mo>,</mo><mtext> </mtext><mn>2</mn></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0147" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mtext> </mtext><mn>0</mn><mo>,</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p61">{−5,−4,−3,−1, 1}</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p63">{−40, −30, −20, 10, 30}</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02_qd02" start="31">
<p class="para" id="fwk-redden-ch01_s01_qs01_p65"><strong class="emphasis bold">Simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p66">−(−10)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p68"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0148" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p70">−(−(−12))</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p72"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0150" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p74"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0152" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p76"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0154" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02_qd03" start="37">
<p class="para" id="fwk-redden-ch01_s01_qs01_p78"><strong class="emphasis bold">Fill in the blank with <, =, or >.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p79">−10 _____ −15</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p81">−101 _____ −100</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p83">−33 _____ 0</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p85">0 _____ −50</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p87">−(−(−2)) _____ −(−3)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0156" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span> _____ <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0157" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0158" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span> _____ <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0159" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0160" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span> _____ <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0161" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02_qd04" start="45">
<p class="para" id="fwk-redden-ch01_s01_qs01_p95"><strong class="emphasis bold">True or False.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0162" display="inline"><mrow><mn>0</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p98"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0163" display="inline"><mrow><mn>5</mn><mo>≤</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0164" display="inline"><mrow><mn>1.0</mn><mover accent="true"><mrow><mn>32</mn></mrow><mo stretchy="true">–</mo></mover></mrow></math></span> is irrational.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p102">0 is a nonnegative number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p104">Any integer is a rational number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p106">The constant <span class="inlineequation"><math xml:id="fwk-redden-ch01_m0165" display="inline"><mi mathvariant="italic">π</mi></math></span> is rational.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02_qd05" start="51">
<p class="para" id="fwk-redden-ch01_s01_qs01_p108"><strong class="emphasis bold">Graph the solution set and give the interval notation equivalent.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0166" display="inline"><mrow><mi>x</mi><mo><</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0167" display="inline"><mrow><mi>x</mi><mo>></mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0169" display="inline"><mrow><mi>x</mi><mo>≥</mo><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0171" display="inline"><mrow><mi>x</mi><mo>≤</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0173" display="inline"><mrow><mo>−</mo><mn>10</mn><mo>≤</mo><mi>x</mi><mo><</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0175" display="inline"><mrow><mn>3</mn><mo><</mo><mi>x</mi><mo>≤</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0177" display="inline"><mrow><mo>−</mo><mn>40</mn><mo><</mo><mi>x</mi><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p123"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0179" display="inline"><mrow><mo>−</mo><mn>12</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p125"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0181" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo><</mo><mn>5</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p127"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0183" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo>≤</mo><mo>−</mo><mn>10</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≥</mo><mo>−</mo><mn>40</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p129"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0185" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo>≤</mo><mn>7</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo><</mo><mn>10</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0187" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo><</mo><mn>1</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>and</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>></mo><mn>3</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p133"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0189" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo><</mo><mo>−</mo><mn>2</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p135"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0191" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo>≤</mo><mn>0</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>4</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0193" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo><</mo><mn>6</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>></mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0195" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>x</mi><mo><</mo><mn>0</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>≤</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02_qd06" start="67">
<p class="para" id="fwk-redden-ch01_s01_qs01_p141"><strong class="emphasis bold">Write an equivalent inequality.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p142">All real numbers less than −15.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p144">All real numbers greater than or equal to −7.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p146">All real numbers less than 6 and greater than zero.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p148">All real numbers less than zero and greater than −5.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p150">All real numbers less than or equal to 5 or greater than 10.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p152">All real numbers between −2 and 2.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd02_qd07" start="73">
<p class="para" id="fwk-redden-ch01_s01_qs01_p154"><strong class="emphasis bold">Determine the inequality given the answers expressed in interval notation.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p155"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0203" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p157"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0205" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>8</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p159"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0207" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p161"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0209" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p163"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0211" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn><mo>,</mo><mn>14</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p165"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0213" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>12</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p167"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0215" display="inline"><mrow><mrow><mo>[</mo><mrow><mn>5</mn><mo>,</mo><mn>25</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p169"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0217" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>30</mn><mo>,</mo><mo>−</mo><mn>10</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p171"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0219" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>[</mo><mrow><mn>3</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p173"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0221" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>19</mn></mrow><mo>]</mo></mrow><mo>∪</mo><mrow><mo>[</mo><mrow><mo>−</mo><mn>12</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p175"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0223" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p177"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0225" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>15</mn></mrow><mo>]</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd03">
<h3 class="title">Part C: Absolute Value</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd03_qd01" start="85">
<p class="para" id="fwk-redden-ch01_s01_qs01_p179"><strong class="emphasis bold">Simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0227" display="inline"><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>9</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0228" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>14</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0229" display="inline"><mrow><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p186"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0230" display="inline"><mrow><mo>−</mo><mrow><mo>|</mo><mn>8</mn><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p188"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0231" display="inline"><mrow><mtext>−</mtext><mrow><mo>|</mo><mrow><mo>−</mo><mfrac><mn>5</mn><mn>8</mn></mfrac></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0233" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mtext>−</mtext><mrow><mo>|</mo><mrow><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0235" display="inline"><mrow><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0236" display="inline"><mrow><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0237" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0238" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mn>10</mn></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0239" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p202"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0240" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mn>20</mn></mrow><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd03_qd02" start="97">
<p class="para" id="fwk-redden-ch01_s01_qs01_p204"><strong class="emphasis bold">Determine the values represented by <em class="emphasis">a</em>.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p205"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0241" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p207"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0243" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p209"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0245" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p211"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0247" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p213"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0249" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p215"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0251" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd04">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s01_qs01_qd04_qd01" start="103">
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p217">Research and discuss the origins and evolution of algebra.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p218">Research and discuss reasons why algebra is a required subject today.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p219">Solution sets to inequalities can be expressed using a graph, interval notation, or set notation. Discuss the merits and drawbacks of each method. Which do you prefer?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch01_s01_qs01_p220">Research and discuss the Fundamental Theorem of Algebra. Illustrate its idea with an example and share your results.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch01_s01_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p03_ans">{1, 5, 9, 13, 17, 21}</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p07_ans">{2}</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p11_ans">{…,−3, −2, −1}</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p15_ans">{13, 15, 17}</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p20_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0122" display="inline"><mrow><mn>3</mn><mo>⋅</mo><mn>5</mn><mo>⋅</mo><mn>13</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p24_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0124" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn><mo>⋅</mo><mn>11</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p28_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0126" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p33_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0129" display="inline"><mrow><mfrac><mn>7</mn><mn>5</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p37_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0133" display="inline"><mrow><mfrac><mn>7</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p41_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0137" display="inline"><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa21_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p45_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m0141" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa23_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch01_s01_qs01_p49_ans">0</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="25">
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa25_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_04/f1a6465aa015203b430567c2f192eca7.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s01_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">