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380 lines
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HTML
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Solving Inequalities</h1>
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<p>Sometimes we need to solve <a href="inequality.html">Inequalities</a> like these:</p>
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<p><br></p>
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr>
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<th align="center"> <div align="center"><b>Symbol</b></div></th>
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<th> <div align="center"><b>Words</b></div></th>
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<th> <div align="center"><b>Example</b></div></th>
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</tr>
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<tr>
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<td class="large" align="center">></td>
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<td>
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<div align="center">greater than</div></td>
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<td class="larger">
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<div align="center">x + 3 <b>></b> 2</div></td>
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</tr>
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<tr>
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<td class="large" align="center"><</td>
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<td>
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<div align="center">less than</div></td>
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<td class="larger">
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<div align="center">7x <b><</b> 28</div></td>
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</tr>
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<tr>
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<td class="large" align="center">≥</td>
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<td>
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<div align="center">greater than or equal to</div></td>
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<td class="larger">
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<div align="center">5 <b>≥</b> x − 1 </div></td>
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</tr>
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<tr>
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<td class="large" align="center">≤</td>
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<td>
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<div align="center">less than or equal to </div></td>
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<td class="larger">
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<div align="center">2y + 1 <b>≤</b> 7</div></td>
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</tr>
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</tbody></table>
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<h2>Solving</h2>
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<p><b>Our aim</b> is to have <span class="large">x</span> (or whatever the variable is) <b>on its own</b> on the left of the inequality sign:</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td align="right">Something like:</td>
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<td width="30"> </td>
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<td class="large">x < 5</td>
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</tr>
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<tr>
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<td align="right">or:</td>
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<td width="30"> </td>
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<td class="large">y ≥ 11</td>
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</tr>
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</tbody></table>
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<p>We call that "solved". </p>
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<div class="example">
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<h3>Example: x + 2 > 12</h3>
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<p>Subtract 2 from both sides:</p>
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<p class="larger" align="center">x + 2 − 2 > 12 − 2</p>
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<p>Simplify:</p>
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<p class="larger" align="center">x > 10</p>
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<p><b>Solved!</b></p>
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</div>
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<h2>How to Solve</h2>
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<p>Solving inequalities is very like <a href="equations-solving.html">solving equations</a> ... we do most of the same things ... </p>
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<p class="larger" align="center">... but we must also pay attention to the <b>direction of the inequality</b>.</p>
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<p class="larger" align="center"><img src="../numbers/images/greater-than-symbol.svg" alt="greater than sign"><br>
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Direction: Which way the arrow "points"<br>
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</p>
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<div class="center80">
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<p>Some things can <b>change the direction</b>! </p>
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<p align="center"><span class="large"><</span> becomes <span class="large">></span></p>
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<p align="center"><span class="large">></span> becomes <span class="large"><</span></p>
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<p align="center"><span class="large">≤</span> becomes <span class="large">≥</span></p>
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<p align="center"><span class="large">≥</span> becomes <span class="large">≤</span></p>
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</div>
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<h2>Safe Things To Do</h2>
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<p>These things <b>do not affect</b> the direction of the inequality:</p>
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<ul>
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<li>Add (or subtract) a number from both sides</li>
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<li>Multiply (or divide) both sides by a <b>positive</b> number</li>
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<li>Simplify a side</li>
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</ul>
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<div class="example">
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<h3>Example: 3x < 7+3</h3>
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<p>We can simplify 7+3 without affecting the inequality:</p>
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<p class="larger" align="center">3x < 10</p>
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</div>
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<p><span class="large">But</span> these things <b> do change the direction</b> of the inequality ("<" becomes ">" for example):</p>
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<ul>
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<div class="bigul">
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<li>Multiply (or divide) both sides by a <b>negative</b> number</li>
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<li>Swapping left and right hand sides</li>
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</div>
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</ul>
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<div class="example">
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<h3>Example: 2y+7 < 12</h3>
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<p>When we swap the left and right hand sides, we must also <b>change the direction of the inequality</b>:</p>
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<p class="larger" align="center">12 <b>></b> 2y+7</p>
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</div>
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<p>Here are the details:</p>
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<h2>Adding or Subtracting a Value</h2>
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<p>We can often solve inequalities by adding (or subtracting) a number from both sides (just as in <a href="introduction.html">Introduction to Algebra</a>), like this:</p>
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<div class="example">
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<h3>Example: x + 3 < 7</h3>
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<p>If we subtract 3 from both sides, we get:</p>
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<p align="center"><span class="larger">x + 3 <b>− 3</b> < 7 <b>− 3</b></span> </p>
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<p class="larger" align="center">x < 4</p>
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<p>And that is our solution: <b>x < 4</b></p>
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<p>In other words, <b>x</b> can be any value less than 4. </p>
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</div>
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<p> </p>
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<h3>What did we do? </h3>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td class="larger" align="right">
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<p class="larger">We went from this: </p>
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<p> </p>
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<p class="larger">To this:</p></td>
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<td align="center"> </td>
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<td align="center"><img src="images/number-line-inequal-xp3lt7.gif" alt="number line inequality x+3 < 7" height="162" width="251"></td>
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<td align="center"> </td>
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<td align="center">
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<p class="larger">x+3 < 7</p>
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<p> </p>
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<p class="larger">x < 4</p></td>
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</tr>
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<tr>
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<td align="center"> </td>
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<td align="center"> </td>
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<td align="center"> </td>
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<td align="center"> </td>
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<td align="center"> </td>
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</tr>
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</tbody></table>
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<p>And that works well for <b>adding</b> and <b>subtracting</b>, because if we add (or subtract) the same amount from both sides, it does not affect the inequality</p>
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<div class="example">
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<p>Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.</p>
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</div>
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<h2>What If I Solve It, But "x" Is On The Right?</h2>
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<p>No matter, just swap sides, but <b>reverse the sign</b> so it still "points at" the correct value!</p>
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<div class="example">
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<h3>Example: 12 < x + 5</h3>
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<p>If we subtract 5 from both sides, we get:</p>
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<p align="center"><span class="larger">12 <b>− 5</b> < x + 5 <b>− 5</b> </span> </p>
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<p class="larger" align="center">7 < x</p>
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<p>That is a solution!</p>
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<p>But it is normal to put "x" on the left hand side ...</p>
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<p align="center">... so let us flip sides (and the inequality sign!):</p>
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<p class="larger" align="center">x > 7</p>
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<p>Do you see how the inequality sign still "points at" the smaller value (7) ?</p>
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<p>And that is our solution: <b>x > 7</b></p>
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</div>
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<p>Note: "x" <b>can</b> be on the right, but people usually like to see it on the left hand side.</p>
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<h2>Multiplying or Dividing by a Value</h2>
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<p>Another thing we do is multiply or divide both sides by a value (just as in <a href="introduction-multiply.html">Algebra - Multiplying</a>).</p>
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<p>But we need to be a bit more careful (as you will see).</p>
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<h3><br>
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Positive Values</h3>
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<p>Everything is fine if we want to multiply or divide by a <b>positive number</b>:</p>
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<div class="example">
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<h3>Example: 3y < 15</h3>
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<p>If we divide both sides by 3 we get:</p>
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<p class="larger" align="center">3y<b>/3</b> < 15<b>/3</b></p>
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<p class="larger" align="center">y < 5</p>
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<p>And that is our solution: <b>y < 5</b></p>
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</div>
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<h3><br>
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Negative Values</h3>
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td><img src="../images/style/warning.gif" alt="warning!" height="34" width="38"></td>
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<td class="larger">When we multiply or divide by a <b>negative number </b><br>
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we must <b>reverse</b> the inequality.</td>
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</tr>
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</tbody></table>
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<br>
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<div class="center80">
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<h3><b>Why?</b> </h3>
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<p>Well, just look at the number line! </p>
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<p align="center">For example, from 3 to 7 is <b>an increase</b>, <br>
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but from −3 to −7 is <b>a decrease. </b></p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td colspan="2" align="center"><img src="images/number-line-inequal-pos-neg.svg" style="max-width:100%" alt="number line -7<-3 and 3<7"></td>
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</tr>
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<tr align="center">
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<td class="large" width="50%">−7 < −3</td>
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<td class="large" width="50%">7 > 3</td>
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</tr>
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</tbody></table>
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<p>See how the inequality sign reverses (from < to >) ?</p>
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</div>
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<p>Let us try an example:</p>
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<div class="example">
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<h3>Example: −2y < −8</h3>
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<p>Let us divide both sides by −2 ... and <b>reverse the inequality</b>!</p>
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<p class="larger" align="center">−2y<b> </b>< −8</p>
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<p class="larger" align="center">−2y<b>/−2</b> <span class="hilite">></span> −8<b>/−2</b></p>
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<p class="larger" align="center">y > 4</p>
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<p>And that is the correct solution: <b>y > 4</b></p>
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</div>
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<p>(Note that I reversed the inequality <b>on the same line</b> I divided by the negative number.)</p>
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<p>So, just remember:</p>
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<div class="def">
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<p>When multiplying or dividing by a negative number, <b>reverse</b> the inequality</p>
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</div>
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<h2>Multiplying or Dividing by Variables</h2>
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<p>Here is another (tricky!) example: </p>
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<div class="example">
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<h3>Example: bx < 3b</h3>
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<p>It seems easy just to divide both sides by <b>b</b>, which gives us:</p>
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<p align="center"><b>x < 3</b></p>
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<p>... but wait ... if <b>b</b> is <b>negative</b> we need to reverse the inequality like this:</p>
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<p align="center"><b>x > 3</b></p>
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<p>But we don't know if b is positive or negative, so <b>we can't answer this one</b>!</p>
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</div>
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<p>To help you understand, imagine replacing <b>b</b> with <b>1 or −1</b> in the example of <b>bx < 3b</b>:</p>
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<ul>
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<li>if <b>b is 1</b>, then the answer is <b>x < 3 </b></li>
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<li>but if <b>b is −1</b>, then we are solving <b>−x < −3</b>, and the answer is <b>x > 3</b></li>
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</ul>
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<p>The answer could be <b>x < 3 </b>or<b> x > 3</b> and we can't choose because we don't know <b>b</b>.</p>
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So:
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<div class="def">
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<p><b>Do not</b> try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).</p>
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</div>
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<h2>A Bigger Example</h2>
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<div class="example">
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<h3>Example: <span class="intbl"><em>x−3</em><strong>2</strong></span> < −5</h3>
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<p>First, let us clear out the "/2" by multiplying both sides by 2.</p>
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<p>Because we are multiplying by a positive number, the inequalities will not change.</p>
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<p class="larger" align="center"><span class="intbl"><em>x−3</em><strong>2</strong></span> <b>×2</b> < −5 <b>×2</b> </p>
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<p class="larger" align="center">x−3 < −10</p>
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<p>Now add 3 to both sides:</p>
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<p class="larger" align="center">x−3 <b>+ 3</b> < −10 +<b> 3</b> </p>
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<p class="larger" align="center">x < −7</p>
|
||
<p>And that is our solution: <b>x < −7</b></p>
|
||
</div>
|
||
<h2>Two Inequalities At Once!</h2>
|
||
<p>How do we solve something with two inequalities at once?</p>
|
||
<div class="example">
|
||
<h3>Example:
|
||
|
||
<div class="center">−2 < <span class="intbl"><em>6−2x</em><strong>3</strong></span> < 4</div></h3>
|
||
<p>First, let us clear out the "/3" by multiplying each part by 3.</p>
|
||
<p>Because we are multiplying by a positive number, the inequalities don't change:</p>
|
||
<p class="larger" align="center">−6 < 6−2x < 12</p>Now subtract 6 from each part:
|
||
|
||
|
||
<p class="larger" align="center">−12 < −2x < 6</p>
|
||
<p>Now divide each part by 2 (a positive number, so again the inequalities don't change):</p>
|
||
<p class="larger" align="center">−6 < −x < 3</p>
|
||
<p>Now multiply each part by −1. Because we are multiplying by a <b>negative</b> number, the inequalities <b>change direction</b>.
|
||
</p>
|
||
<p class="larger" align="center"></p>
|
||
<p class="larger" align="center">6 <span class="hilite">></span> x <span class="hilite">></span> −3</p>
|
||
|
||
<p>And that is the solution!</p>
|
||
<p>But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):</p>
|
||
<p class="large" align="center">−3 <span class="hilite"><</span> x <span class="hilite"><</span> 6</p>
|
||
</div>
|
||
<p> </p>
|
||
<h2>Summary</h2>
|
||
<ul class="larger">
|
||
|
||
<li>Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.</li>
|
||
<li>But these things will change direction of the inequality:
|
||
|
||
|
||
<ul>
|
||
<li>Multiplying or dividing both sides by a <b>negative</b> number</li>
|
||
<li>Swapping left and right hand sides </li>
|
||
</ul>
|
||
</li>
|
||
<li>Don't multiply or divide by a <b>variable</b> (unless you know it is always positive or always negative)</li>
|
||
</ul>
|
||
|
||
<p>
|
||
</p>
|
||
<div class="questions">
|
||
|
||
|
||
<script type="text/javascript">getQ(447,448,303,304,449,450, 1222, 1223, 1224, 1225);</script>
|
||
|
||
|
||
</div>
|
||
<div class="related">
|
||
<a href="../equal-less-greater.html">Less Than or Greater Than</a>
|
||
<a href="inequality.html">Inequalities</a>
|
||
<a href="inequality-questions-solving.html">Solving Inequality Word Questions</a>
|
||
<a href="graphing-linear-inequalities.html">Graphing Linear Inequalities</a>
|
||
<a href="../data/inequality-grapher.html">Inequality Grapher</a>
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