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348 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Properties of Inequalities</h1>
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<p class="center"><strong>Inequality</strong>
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tells us about the <b>relative size</b> of two values.</p>
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<p class="center"><i>(You might like to read a gentle <a href="inequality.html">Introduction to Inequalities</a> first)</i></p>
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<h2>The 4 Inequalities</h2>
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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> <div class="center"><b>Symbol</b></div></th>
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<th> <div class="center"><b>Words</b></div></th>
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<th> <div class="center"><b>Example</b></div></th>
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</tr>
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<tr>
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<td class="larger">
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<div class="center larger">></div></td>
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<td>
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<div class="center">greater than</div></td>
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<td>
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<div class="center">x+3 > 2</div></td>
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</tr>
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<tr>
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<td class="larger">
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<div class="center larger"><</div></td>
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<td>
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<div class="center">less than</div></td>
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<td>
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<div class="center">7x < 28</div></td>
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</tr>
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<tr>
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<td class="larger">
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<div class="center larger">≥</div></td>
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<td>
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<div class="center">greater than or equal to</div></td>
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<td>
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<div class="center">5 ≥ x−1 </div></td>
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</tr>
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<tr>
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<td>
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<div class="center larger">≤</div></td>
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<td>
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<div class="center">less than or equal to </div></td>
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<td>
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<div class="center">2y+1 ≤ 7</div></td>
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</tr>
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<tr height="6">
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<td><br>
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</td>
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<td><br>
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</td>
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<td><br>
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</td>
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</tr>
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</tbody></table>
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<p class="center"><img src="../numbers/images/greater-than-symbol.svg" alt="greater than sign" height="104" width="236"></p>
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<p class="center large">The symbol "points at" the smaller value</p>
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<h2>Properties</h2>
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<p>Inequalities have properties ... all with special names!</p>
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<p>Here we list each one, with examples.</p>
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<p>Note: the values <b>a</b>, <b>b</b> and <b>c</b> we use below are <a href="../numbers/real-numbers.html">Real Numbers</a>.</p>
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<p> </p>
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<h2>Transitive Property</h2>
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<p>When we link up inequalities in order, we can "jump over" the middle inequality.</p>
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<p class="center"><img src="images/inequality-transitive.svg" alt="Transitive Property" height="73" width="298"></p>
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<p class="center"><span class="larger">If a < b <b>and</b> b < c, then a < c</span></p>
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<p>Likewise:</p>
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<p class="center larger">If a > b <b>and</b> b > c, then a > c</p>
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<div class="example">
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<h3>Example:</h3>
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<ul>
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<li>If Alex is older than Billy and</li>
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<li>Billy is older than Carol,</li>
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</ul>
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<p>then Alex must be older than Carol also!</p>
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</div>
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<h2>Reversal Property</h2>
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<p>We can swap <b>a</b> and <b>b</b> over, if we make sure the symbol still "points at" the smaller value.</p>
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<ul>
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<li>If a > b then b < a</li>
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<li>If a < b then b > a</li>
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</ul>
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<div class="example">
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<p>Example: Alex is older than Billy, so Billy is younger than Alex</p>
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</div>
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<h2>Law of Trichotomy</h2>
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<p>The "Law of Trichotomy" says that<b> only one</b> of the following is true:</p>
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<div class="center"><img src="images/inequality-trichotomy.svg" alt="Trichotomy Property" height="38" width="358"> </div>
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<p>It makes sense, right? <b>a</b> must be either <b>less than b</b> or <b>equal to b</b> or <b>greater than b</b>. It must be one of those, and only one of those.</p>
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<div class="example">
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<h3>Example: Alex Has More Money Than Billy</h3>
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<p>We could write it like this:</p>
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<p class="center large">a > b</p>
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<p>So we also know that:</p>
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<ul>
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<li>Alex does <b>not</b> have <b>less</b> money than Billy (not a<b)</li>
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<li>Alex does <b>not</b> have <b>the same amount</b> of money as Billy (not a=b)</li>
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</ul>
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<p>(Of course!)</p>
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</div>
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<p> </p>
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<h2>Addition and Subtraction</h2>
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<p>Adding <b>c</b> to both sides of an inequality just <b>shifts everything along</b>, and the inequality stays the same.</p>
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<p class="center"><img src="images/inequality-addition.gif" alt="Addition Property" height="93" width="301"></p>
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<p class="center"><span class="larger">If a < b, then a <b>+ c</b> < b <b>+ c</b> </span></p>
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<div class="example">
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<h3>Example: Alex has less money than Billy.</h3>
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<p>If both Alex and Billy get $3 more, then Alex will still have less money than Billy.</p>
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</div>
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<p>Likewise:</p>
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<ul>
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<li>If a < b, then a − c < b − c</li>
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<li>If a > b, then a + c > b + c, and</li>
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<li>If a > b, then a − c > b − c</li>
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</ul>
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<p>So adding (or subtracting) the same value to both a and b <b>will not change</b> the inequality</p>
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<p> </p>
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<h2>Multiplication and Division</h2>
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<p>When we multiply both a and b by a <b>positive number</b>, the inequality <b>stays the same</b>.</p>
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<p>But when we multiply both a and b by a <b>negative number</b>, the inequality <b>swaps over</b>!</p>
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<p class="center"><img src="images/inequality-multiplication.gif" alt="Multiplication Property" height="121" width="387"><br>
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<span class="larger">Notice that <b>a<b</b> becomes <b>b<a</b> after multiplying by (-2)</span><br>
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But the inequality stays the same when multiplying by +3</p>
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<p>Here are the rules:</p>
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<ul>
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<li>If a < b, and <b>c is positive</b>, then <b>ac < bc</b></li>
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<li>If a < b, and <b>c is negative</b>, then <b>ac > bc</b> (inequality swaps over!)</li>
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</ul>
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<p>A "positive" example:</p>
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<div class="example">
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<p>Example: Alex's score of 3 is <b>lower than</b> Billy's score of 7.</p>
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<p class="center large">a < b</p>
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<p>If both Alex and Billy manage to <b>double</b> their scores (×2), Alex's score will still be lower than Billy's score.</p>
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<p class="center large">2a < 2b</p>
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</div>
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<p>But when multiplying by a negative the opposite happens:</p>
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<div class="example">
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<p>But if the scores become <b>minuses</b>, then Alex <b>loses 3</b> points and Billy <b>loses 7</b> points</p>
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<p>So Alex has now done <b>better</b> than Billy!</p>
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<p class="center large">−a > −b</p>
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</div>
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<div class="center80">
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<h3><b>Why does multiplying by a negative reverse the sign?</b></h3>
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<p>Well, just look at the number line!</p>
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<p>For example, from −3 to −7 is <b>a decrease</b>, but from 3 to 7 is <b>an increase</b>.</p>
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<p class="center"><img src="images/number-line-inequal-pos-neg.svg" alt="number line -7<-3 and 3<7" height="93" width="592"></p>
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<div class="tbl">
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<div class="row"><span class="left">Notice that</span><span class="right">−7 < −3</span></div>
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<div class="row"><span class="left">but</span><span class="right">+7 > +3</span></div>
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</div>
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<p>So the inequality sign reverses (from < to >)</p>
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</div>
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<h2>Additive Inverse</h2>
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<p>As we just saw, putting minuses in front of a and b <b>changes the direction</b> of the inequality. This is called the "Additive Inverse":</p>
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<ul>
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<li>If a < b then −a > −b</li>
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<li>If a > b then −a < −b</li>
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</ul>
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<p>This is really the same as multiplying by (-1), and that is why it changes direction.</p>
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<div class="example">
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<p>Example: Alex has more money than Billy, and so Alex is ahead.</p>
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<p>But a new law says "all your money is now a <b>debt</b> you must repay with hard work"</p>
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<p>So now Alex is worse off than Billy.</p>
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</div>
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<h2>Multiplicative Inverse</h2>
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<p>Taking the <a href="reciprocal.html">reciprocal</a> (1/value) of both a and b <b>can change the direction</b> of the inequality.</p>
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<p class="center"><img src="images/inequality-multiplicative-inverse.svg" alt="Multiplicative Inverse" height="" width=""></p>
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<p>When a and b are <b>both positive</b> or <b>both negative</b>:</p>
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<ul>
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<li>If a < b then 1/a <span class="hilite">></span> 1/b</li>
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<li>If a > b then 1/a <span class="hilite"><</span> 1/b</li>
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</ul>
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<div class="example">
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<h3>Example: Alex and Billy both complete a journey of 12 kilometers.</h3>
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<p>Alex runs at <b>6 km/h</b> and Billy walks at <b>4 km/h</b>.</p>
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<p>Alex’s speed is greater than Billy’s speed</p>
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<p class="center larger">6 > 4</p>
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<p>But Alex’s time is less than Billy’s time:</p>
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<p class="center">12/6 < 12/4</p>
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<p class="center larger">2 hours < 3 hours</p>
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</div>
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<p>But when either <b>a or b is negative</b> (not both) the direction stays the same:</p>
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<ul>
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<li>If a < b then 1/a < 1/b</li>
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<li>If a > b then 1/a > 1/b</li>
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</ul>
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<div class="example">
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<h3>Example: a = +7 and b = −3</h3>
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<p>
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a > b, and one of them is negative, so:
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</p>
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<p class="so"><span class="intbl"><em>1</em><strong>+7</strong></span> > <span class="intbl"><em>1</em><strong>−3</strong></span></p>
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<!-- When a = 7 and b = -3 then 1/7 > 1/-3 -->
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<p class="so"><span class="intbl"><em>1</em><strong>7</strong></span> > −<span class="intbl"><em>1</em><strong>3</strong></span></p></div>
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<h2>Non-Negative Property of Squares</h2>
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<p>A square of a number is greater than or equal to zero:</p>
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<p class="center larger">a<sup>2</sup> ≥ 0</p>
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<div class="example">
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<h3>Example:</h3>
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<ul>
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<li>(3)<sup>2</sup> = 9</li>
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<li>(−3)<sup>2</sup> = 9</li>
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<li>(0)<sup>2</sup> = 0</li>
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</ul>
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<p>Always greater than (or equal to) zero</p>
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</div>
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<h2>Square Root Property</h2>
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<p>Taking a square root will not change the inequality <i>(but only when both a and b are greater than or equal to zero)</i>.</p>
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<p class="center"><span class="larger">If a ≤ b then √a ≤ √b<br>
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</span>(for a,b ≥ 0)</p>
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<div class="example">
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<h3>Example: a=4, b=9</h3>
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<ul>
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<li>4 ≤ 9 so √4 ≤ √9</li>
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</ul>
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</div>
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<p> </p>
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<div class="questions">2064, 2065, 2066, 2067, 445, 446, 2320, 2321, 2322, 2323</div>
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<div class="related">
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<a href="inequality.html">Inequalities</a>
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||
<a href="inequality-solving.html">Solving Inequalities</a>
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||
<a href="inequality-questions-solving.html">Solving Inequality Word Questions</a>
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<a href="../equal-less-greater.html">Less Than or Greater Than</a>
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</div>
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