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379 lines
15 KiB
HTML
379 lines
15 KiB
HTML
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<div class="video">lo1kvxu-Dc8</div>
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<h1 id="title" align="center">Equal, Greater or Less Than</h1>
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<p class="center"><i>As well as the familiar equals sign (=) it is also very useful to show if something is not equal to (≠) greater than
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(>) or less than (<)</i></p>
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<p><b>These are the important signs to know</b>:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td class="huge style1 style2" align="center">=</td>
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<td style="text-align:center;">
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<p>When two values are equal<br>
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we
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use the "equals" sign</p></td>
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<td>example: <b>2+2 = 4</b></td>
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</tr>
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<tr>
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<td class="huge style1 style2" align="center">≠</td>
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<td style="text-align:center;">
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<p>When two values are definitely <b>not</b> equal<br>
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we
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use the "not equal to" sign</p></td>
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<td>example: <b>2+2 ≠ 9</b></td>
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</tr>
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<tr>
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<td class="huge style2" align="center"><</td>
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<td style="text-align:center;">
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<p>When one value is smaller than another<br>
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we
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use a "less than" sign</p></td>
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<td>example: <b>3 < 5</b></td>
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</tr>
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<tr>
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<td class="huge style2" align="center">></td>
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<td style="text-align:center;">
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<p>When one value is bigger than another<br>
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we
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use a "greater than" sign</p></td>
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<td>example: <b>9 > 6</b></td>
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</tr>
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</tbody></table>
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</div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/sage-greater-than.svg" alt="greater than sign"></p>
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<h2>Less Than and Greater Than</h2>
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<p>The "less than" sign and the "greater than" sign look like a "V" on its side, don't they?</p>
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<p>To remember which way around the "<" and ">" signs go, just remember:</p>
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<ul>
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<li><span class="large">BIG > small</span></li>
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<li><span class="large">small < BIG</span></li>
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</ul>
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The "small" end always points to the smaller number, like this:
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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">Greater Than Symbol: <b>BIG > small</b></p>
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<p> </p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large">10 > 5</p>
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<p class="center"><i>"10 is <b>greater than</b> 5"</i></p>
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<p align="left">Or the other way around:</p>
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<p class="center large">5 < 10</p>
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<p class="center"><i>"5 is <b>less than</b> 10"</i></p>
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<p align="left"> </p>
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<p align="left">Do you see how the symbol "points at" the smaller value?</p>
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</div>
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<h2>... Or Equal To ...</h2>
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<p>Sometimes we know a value is smaller, but <b>may also be equal to</b>!</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="measure/images/4cup-jug.gif" alt="jug" height="163" width="179"></p>
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<h3>Example, a jug can hold up to 4 cups of water.</h3>
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<p>So how much water is in it?</p>
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<p>It could be 4 cups or it could be less than 4 cups: So until we measure it, all we can say is "less than <b>or equal to</b>" 4 cups.</p>
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</div>
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<p>To show this, we add an extra line at the bottom of the "less than" or "greater than" symbol like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>The "less than <b>or equal to</b>" sign:</p> </td>
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<td style="text-align:right; width:20px;"> </td>
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<td style="text-align:center;"><span class="huge">≤</span></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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<td style="text-align:right;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:right;">
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<p align="right">The "greater than <b>or equal to</b>" sign:</p> </td>
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<td style="text-align:right;"> </td>
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<td style="text-align:center;"><span class="huge">≥</span></td>
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</tr>
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</tbody></table>
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<h2>All The Symbols</h2>
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<p>Here is a summary of all the symbols:</p>
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<div class="simple">
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<table align="center" cellspacing="5" border="0">
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<tbody>
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<tr>
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<th>
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<div class="center"><b>Symbol</b></div> </th>
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<th>
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<div class="center"><b>Words</b></div> </th>
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<th>
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<div class="center"><b>Example Use</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">=</div> </td>
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<td>
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<div class="center">equals</div> </td>
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<td>
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<div class="center">1 + 1 = 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">≠</div> </td>
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<td>
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<div class="center">not equal to</div> </td>
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<td>
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<div class="center"> 1 + 1 ≠ 1</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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<tr>
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<td class="larger">
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<div class="center">></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">5 > 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"><</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">7 < 9</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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<tr>
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<td class="larger">
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<div class="center">≥</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">marbles ≥ 1 </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">≤</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">dogs ≤ 3</div> </td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<h2>Why Use Them?</h2>
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<p>Because there are things we <b>do not know <i>exactly ...</i></b></p>
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<div class="center">
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... but can still <b>say <i>something</i> about</b>.</div>
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<p>So we have ways of saying what we <b>do</b> know (which may be useful!)</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/marbles.svg" alt="marbles"></p>
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<h3>Example: John had 10 marbles, but lost some. How many has he now?</h3>
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<p>Answer: He must have <b>less than</b> 10:</p>
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<p class="center large">Marbles <b><</b> 10</p>
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<p> </p>
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<p>If John still has some marbles we can also say he has <b>greater than zero</b> marbles:</p>
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<p class="center"><span class="large">Marbles <b>></b> 0</span></p>
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<p> </p>
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<p>But if we thought John <b>could have</b> lost <b>all</b> his marbles we would say</p>
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<p class="center"><span class="large">Marbles <b>≥</b> 0</span></p>
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<p>In other words, the number of marbles is greater than <b>or equal to</b> zero.</p>
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</div>
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<h2>Combining</h2>
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<p>We can sometimes say two (or more) things on the one line:</p>
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<div class="example">
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<h3>Example: Becky starts with $10, buys something and says "I got change, too". How much did she spend?</h3>
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<p>Answer: Something greater than $0 and less than $10 (but NOT $0 or $10):</p>
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<p class="center larger">"What Becky Spends" > $0<br>
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"What Becky Spends" < $10</p>
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<p>This can be written down in just one line:</p>
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<p class="center larger"><b>$0 < "What Becky Spends" < $10</b></p>
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<p>That says that $0 is less than "What Becky Spends" (in other words "What Becky Spends" is greater than $0) and what Becky Spends is also less than $10.</p>
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<p><b>Notice that ">" was flipped over to "<" when we put it <i>before</i> what Becky spends.</b> Always make sure
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the <b>small end points to the small value</b>.</p>
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</div>
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<h2>Changing Sides</h2>
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<p>We saw in that previous example that when we change sides we flipped the symbol as well.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">This:</td>
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<td style="width:20px;"> </td>
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<td class="larger" nowrap="nowrap">Becky Spends > $0</td>
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<td style="width:20px;"> </td>
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<td>(Becky spends greater than $0) </td>
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</tr>
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<tr>
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<td style="text-align:right;">is the same as this:</td>
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<td> </td>
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<td class="larger" nowrap="nowrap">$0 < Becky Spends</td>
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<td> </td>
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<td>($0 is less than what Becky spends)</td>
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</tr>
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</tbody></table><br>
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<div class="def">
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<p class="center larger">Just make sure the small end points to the small value!</p>
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</div>
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<p> </p>
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<p>Here is another example using <span class="larger">"≥</span>" and <span class="larger">"≤</span>":</p>
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<div class="example">
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<h3>Example: Becky has $10 and she is going shopping. How much will she <b>spend</b> (without using credit)?</h3>
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<p>Answer: Something greater than, or possibly equal to, $0 and less than, or possibly equal to, $10:</p>
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<p class="center larger">Becky Spends ≥ $0<br>
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Becky Spends ≤ $10</p>
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<p>This can be written down in just one line:</p>
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<p class="center larger"><b>$0 ≤ Becky Spends ≤ $10</b></p>
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<p> </p>
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</div>
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<h2>A Long Example: Cutting Rope</h2>
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<p>Here is an interesting example I thought of:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/rope.svg" alt="rope"></p>
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<h3>Example: Sam cuts a 10m rope into two. How long is the longer piece? How long is the shorter piece?</h3>
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<p>Answer: Let us call the <b>longer</b> length of rope "<b>L</b>", and the <b>shorter</b> length "<b>S</b>"</p>
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<p><b>L</b> must be greater than 0m (otherwise it isn't a piece of rope), and also less than 10m:</p>
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<p class="center large">L > 0<br>
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L < 10</p>
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<p>So:</p>
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<p class="center large"><b>0 < L < 10</b></p>
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<p>That says that <b>L</b> (the Longer length of rope) is between 0 and 10 (but not 0 or 10)</p>
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<p> </p>
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<p>The same thing can be said about the shorter length "<b>S</b>":</p>
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<p class="center large"><b>0 < S < 10</b></p>
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<p> </p>
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<p>But I did say there was a "shorter" and "longer" length, so we also know:</p>
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<p class="center large"><b>S < L</b></p>
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<p><i>(Do you see how neat mathematics is? Instead of saying "the shorter length is less than the longer length", we can just write "<b>S < L</b>")</i></p>
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<p> </p>
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<p>We can combine all of that like this:</p>
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<p class="center large"><b>0 < S < L < 10</b></p>
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<p>That says a lot:</p>
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<p><i>0 is less that the short length, the short length is less than the long length, the long length is less than 10.</i></p>
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<p>Reading "backwards" we can also see:</p>
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<p><i>10 is greater than the long length, the long length is greater than the short length, the short length is greater than 0.</i></p>
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<p>It also lets us see that "S" is less than 10 (by "jumping over" the "L"), and even that 0<10 (which we know anyway), all in one statement.</p>
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</div><br>
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<div class="example">
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<p>NOW, I have one more trick. If Sam tried really hard he might be able to cut the rope EXACTLY in half, so each half is 5m, but we know he didn't because we said there was a "shorter" and "longer" length, so we also know:</p>
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<p class="center large">S<5</p>
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<p>and</p>
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<p class="center large">L>5</p>
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<p>We can put that into our very neat statement here:</p>
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<p class="center large">0 < S < 5 < L < 10</p>
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<p>And IF we thought the two lengths MIGHT be exactly 5 we could change that to</p>
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<p class="center large">0 < S ≤ 5 ≤ L < 10</p>
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</div>
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<h2>An Example Using Algebra</h2>
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<p>OK, this example may be complicated if you don't know <a href="algebra/index.html">Algebra</a>, but I thought you might like to see it anyway:</p>
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<div class="example">
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<h3>Example: What is x+3, when we know that x is greater than 11?</h3>
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<p class="center larger"><i>If</i> x > 11 , <i>then</i> x+3 > 14</p>
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<p>(Imagine that "x" is the number of people at your party. If there are more than 11 people at your party, and 3 more arrive, then there must be more than 14 people at your party now.)</p>
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</div><p> </p>
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<a id="q"></a>
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<div class="questions">5250, 5251, 5252, 5253, 5254, 5255, 5256, 5257, 5258, 5259</div>
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<div class="related">
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<a href="number-line.html">Number Line</a>
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<a href="algebra/index.html">Algebra Index</a>
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