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<title>Absolute Value in Algebra</title>
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<h1 align="center">Absolute Value in Algebra</h1>
<h3>Absolute Value means ...</h3>
<p class="center">... <b>how far</b> a number is from zero:</p>
<p class="center"><img src="../numbers/images/absolute-value.svg" alt="absolute value 6 either way" style="max-width:100%" /></p>
<p class="center">&quot;6&quot; is 6 away from zero, <br />
and &quot;&minus;6&quot; is <b>also</b> 6 away from zero. </p>
<p class="center">So the absolute value of 6 is <b>6</b>, <br />
and the absolute value of &minus;6 is also <b>6</b></p>
<p>&nbsp;</p>
<h3>Absolute Value Symbol</h3>
<p>To show we want the absolute value we put &quot;|&quot; marks either side (called &quot;bars&quot;), like these examples:</p>
<table border="0" align="center">
<tr align="center">
<td width="180" class="large">|&minus;5| = 5</td>
<td width="180" class="large">|7| = 7</td>
</tr>
</table>
<br />
<table border="0" align="center">
<tr>
<td><img src="images/key-vertical-bar.jpg" alt="vertical bar" width="84" height="77" /></td>
<td>The &quot;|&quot; can be found just above the enter key on most keyboards. </td>
</tr>
</table>
<h2>More Formal</h2>
<p>More formally we have: </p>
<p align="center"><img src="images/absolute-value-formal.svg" alt="Absolute Value" /></p>
<p>Which says the absolute value of x equals:</p>
<ul>
<li><span class="large">x</span> when x is greater than zero</li>
<li><span class="large">0</span> when x equals 0</li>
<li><span class="large">&minus;x</span> when x is less than zero (this &quot;flips&quot; the number back to positive)</li>
</ul>
<p>So when a number is positive or zero we leave it alone, when it is negative we change it to positive using &minus;x. </p>
<div class="example">
<h3>Example: what is <span class="large">|&minus;17|</span> ?</h3>
<p>Well, it is less than zero, so we need to calculate &quot;&minus;x&quot;:</p>
<p align="center">&minus; ( &minus;17 ) = +<b>17</b></p>
<p>(Because <a href="../positive-negative-integers.html">two minuses make a plus</a>)</p>
</div>
<h2>Useful Properties</h2>
<p>Here are some properties of absolute values that can be useful:</p>
<ul class="larger">
<li> <span class="large">|a| &ge; 0 </span>always!
<p>That makes sense ... <span class="large">|a|</span> can never be less than zero.</p>
</li>
<li> <span class="large">|a| = &radic;(a<sup>2</sup>)</span>
<p>Squaring <span class="large">a</span> makes it positive or zero (for <span class="large">a</span> as a Real Number). Then taking the square root will &quot;undo&quot; the squaring, but leave it positive or zero. </p>
</li>
<li> <span class="large">|a &times; b| = |a| &times; |b|</span>
<p>Means these are the same:</p>
<ul>
<li>the absolute value of (a times b), and</li>
<li>(the absolute value of a) times (the absolute value of b)</li>
</ul>
<p>Which can also be useful when solving</p>
</li>
<li> <span class="large">|u| = a </span> is the same as <span class="large">u = &plusmn;a </span> and vice versa
<p>Which is often the key to solving most absolute value questions.</p>
</li>
</ul> <div class="example">
<h3>Example: Solve <span class="large">|x+2| = 5</span></h3>
<p>Using <b>&quot;|u| = a is the same as u = &plusmn;a</b>&quot;:</p>
<div class="tbl">
<div class="row"><span class="left">this:</span><span class="right">|x+2| = 5</span></div>
<div class="row"><span class="left">is the same as this:</span><span class="right">x+2 = &plusmn;5</span></div>
</div>
<p>&nbsp;</p>
<p align="center">Which has two solutions:</p>
<table width="80%" border="0" align="center">
<tr align="center">
<td>x+2 = &minus;5</td>
<td>x+2 = +5</td>
</tr>
<tr align="center">
<td><b>x = &minus;7</b></td>
<td><b>x = 3</b></td>
</tr>
</table>
</div>
<h2>Graphically</h2>
<p>Let us graph that example:</p>
<p align="center"><span class="larger">|x+2| = 5</span></p>
<p>It is easier to graph when we have an &quot;=0&quot; equation, so subtract 5 from both sides:</p>
<p align="center"><span class="larger">|x+2| &minus; 5 = 0</span></p>
<p>So now we can plot <b>y=|x+2|&minus;5</b> and find where it equals zero.</p>
<p><a href="../data/function-grapherc523.html?func1=abs(x+2)-5&amp;xmin=-12&amp;xmax=12&amp;ymin=-8&amp;ymax=8">Here is the plot of y=|x+2|&minus;5</a>, but just for fun let's <b>make the graph by shifting it around</b>:</p>
<table border="0" align="center">
<tr>
<td colspan="3" align="center"><img src="images/absolute-value-x2m5.svg" alt="|x+2| - 5 = 0" style="max-width:100%" /></td>
</tr>
<tr>
<td width="33%" align="center">Start with <b>y=|x|</b> </td>
<td width="33%" align="center">then shift it left to make<br>
it <b>y=|x+2|</b></td>
<td width="33%" align="center">then shift it down to make<br>
it <b>y=|x+2|&minus;5</b></td>
</tr>
</table>
<p align="center">And the two solutions (circled) are <span class="larger">&minus;7</span> and <span class="larger">+3</span>.</p>
<h2>Absolute Value Inequalities</h2>
<p>Mixing Absolute Values and <a href="inequality-solving.html">Inequalites</a> needs a little care!</p>
<p>There are 4 inequalities:</p>
<div class="simple">
<table border="0" align="center">
<tr align="center">
<th width="100" class="large">&lt;</th>
<th width="100" class="large">&le;</th>
<th width="10" class="large">&nbsp;</th>
<th width="100" class="large">&gt;</th>
<th width="100" class="large">&ge;</th>
</tr>
<tr align="center">
<td>less than</td>
<td>less than
<br /> or equal to </td>
<td>&nbsp;</td>
<td>greater than</td>
<td>greater than
<br /> or equal to</td>
</tr>
</table>
</div>
<h2>Less Than, Less Than or Equal To</h2>
<p>With &quot;<span class="large">&lt;</span>&quot; and &quot;<span class="large">&le;</span>&quot; we get <b>one interval</b> centered on zero:</p>
<div class="example">
<h3>Example: Solve |x| &lt; 3</h3>
<p>This means the distance from <b>x</b> to zero must be less than 3:</p>
<p align="center"><img src="images/absolute-value-xlt3.gif" alt="-3 to 3" width="249" height="95" /></p>
<p align="center">Everything in between (but not including) -3 and 3</p>
<p>It can be rewritten as:</p>
<p align="center" class="large">&minus;3 &lt; x &lt; 3</p>
<p align="center">&nbsp;</p>
<p align="center">As an <a href="../sets/intervals.html">interval</a> it can be written as:</p>
<p align="center"><span class="large"> (&minus;3, 3)</span></p>
</div>
<p>The same thing works for &quot;Less Than or Equal To&quot;:</p>
<div class="example">
<h3>Example: Solve |x| &le; 3</h3>
<p>Everything in between <b>and including</b> -3 and 3</p>
<p>It can be rewritten as:</p>
<p align="center" class="large">&minus;3 &le; x &le; 3</p>
<p align="center">As an <a href="../sets/intervals.html">interval</a> it can be written as:</p>
<p align="center"><span class="large"> [&minus;3, 3]</span></p>
</div>
<p>How about a bigger example? </p>
<div class="example">
<h3>Example: Solve |3x-6| &le; 12</h3>
<p>Rewrite it as:</p>
<p align="center" class="large">&minus;12 &le; 3x&minus;6 &le; 12</p>
<p>Add 6:</p>
<p align="center" class="large">&minus;6 &le; 3x &le; 18</p>
<p align="center"></p>
<p>Lastly, multiply by (1/3). Because we are multiplying by a positive number, the inequalities will not change:</p>
<p align="center" class="large">&minus;2 &le; x &le; 6</p>
<p align="center"></p>
<p align="center"><b>Done!</b></p>
<p align="center"></p>
<p align="center">As an <a href="../sets/intervals.html">interval</a> it can be written as:</p>
<p align="center"><span class="large"> [&minus;2, 6]</span></p>
</div>
<h2>Greater Than, Greater Than or Equal To</h2>
<p>This is different ... we get <b>two separate intervals</b>:</p>
<div class="example">
<h3>Example: Solve |x| &gt; 3</h3>
<p>It looks like this:</p>
<p align="center"><img src="images/absolute-value-xgt3.gif" alt="|x| &gt; 3" width="348" height="61" /></p>
<p align="center">Up to -3 <b>or</b> from 3 onwards</p>
<p>It can be rewritten as</p>
<p align="center" class="large">x &lt; &minus;3 &nbsp; <i>or</i> &nbsp; x &gt; 3</p>
<p align="center">&nbsp;</p>
<p align="center">As an <a href="../sets/intervals.html">interval</a> it can be written as:</p>
<p align="center"><span class="large"> (&minus;&infin;, &minus;3) U </span><span class="large">(3, +&infin;)</span></p>
</div>
<div class="center80">
<p>Careful! <b>Do not</b> write it as </p>
<p align="center"><span class="large">&minus;3 &gt; x &gt; 3</span> &nbsp; &nbsp; <img src="../images/style/no.svg" alt="no!" /></p>
<p align="center">&quot;x&quot; cannot be less than -3 <b>and</b> greater than 3 at the same time</p>
<p>It is really:</p>
<p align="center"><span class="large">x &lt; &minus;3 &nbsp; <i>or</i> &nbsp; x &gt; 3</span> &nbsp; &nbsp; <img src="../images/style/yes.svg" alt="yes" /></p>
<p align="center">&quot;x&quot; is less than &minus;3 <b>or</b> greater than 3</p>
</div>
<p>&nbsp;</p>
<p>The same thing works for &quot;Greater Than or Equal To&quot;:</p>
<div class="example">
<h3>Example: Solve |x| &ge; 3</h3>
<p>Can be rewritten as</p>
<p align="center" class="large">x &le; &minus;3 &nbsp; <i>or</i> &nbsp; x &ge; 3</p>
<p align="center">&nbsp;</p>
<p align="center">As an <a href="../sets/intervals.html">interval</a> it can be written as:</p>
<p align="center"> <span class="large"> (&minus;&infin;, &minus;3] U </span><span class="large">[3, +&infin;)</span></p>
</div>
<p>&nbsp;</p>
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<div class="related"> <a href="../numbers/absolute-value.html">Absolute Value (Introduction)</a> <a href="../sets/function-absolute-value.html">Absolute Value Function</a> </div>
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