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394 lines
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Intervals</h1>
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<p align="center" class="larger">Interval: all the numbers between two given numbers.</p>
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<div class="example">
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<p>Example: all the numbers between 1 and 6 is an <b>interval</b></p>
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</div>
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<h2>All The Numbers?</h2>
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<p>Yes. All the <a href="../numbers/real-numbers.html">Real Numbers</a> that lie between those 2 values.</p>
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<div class="example">
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<h3>Example: the interval 2 to 4 includes numbers such as:</h3>
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<table border="0" align="center">
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<tr>
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<td width="60" align="center"><b>2.1</b></td>
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<td width="60" align="center"><b>2.1111</b></td>
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<td width="60" align="center"><b>2.5</b></td>
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<td width="60" align="center"><b>2.75</b></td>
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<td width="60" align="center"><b>2.80001</b></td>
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<td width="60" align="center">
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<div class="times"><b>π</b></div>
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</td>
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<td width="60" align="center"><b><span class="frac"><sup>7</sup>/<sub>2</sub></span></b></td>
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<td width="60" align="center"><b>3.7937</b></td>
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</tr>
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</table>
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<p>And lots more!</p>
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</div>
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<h2>Including the Numbers at Each End?</h2>
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<p>Ahh ... maybe yes, maybe no ... we need to say! </p>
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<div class="example">
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<h3>Example: "boxes up to 20 kg in mass are allowed"</h3>
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<p>If your box is <b>exactly 20 kg</b> ... will that be allowed or not?</p>
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<p>It isn't really clear.</p>
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</div>
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<p>Let's see how to be precise about this in each of three popular methods:</p>
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<ul>
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<li>Inequalities </li>
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<li>The Number Line </li>
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<li>Interval Notation</li>
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</ul>
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<h2>Inequalities</h2>
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<p>With <a href="../algebra/inequality.html">Inequalities</a> we use:</p>
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<ul>
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<li><span class="large">></span> greater than</li>
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<li><span class="large">≥</span> greater than or equal to</li>
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<li><span class="large"><</span> less than</li>
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<li><span class="large">≤</span> less than or equal to</li>
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</ul>
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<p>Like this:</p>
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<div class="example">
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<h3>Example: <span class="large">x ≤ 20</span></h3>
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<p>Says: "x less than or equal to 20"</p>
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<p>And means: up to <b>and including</b> 20</p>
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</div>
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<h2>Interval Notation</h2>
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<p>In "Interval Notation" we just write the beginning and ending numbers of the interval, and use:</p>
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<ul>
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<li><span class="large"> [ ]</span> a square bracket when we want to <b>include</b> the end value, or</li>
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<li><span class="large">( )</span> a round bracket when we <b>don't</b></li>
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</ul>
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<p>Like this:</p>
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<p align="center"><img src="images/interval-notation.svg" alt="Interval Notation" /></p>
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<div class="example">
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<h3>Example: (5, 12]</h3>
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<p>Means from 5 to 12, do <b>not</b> include 5, but <b>do</b> include 12</p>
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</div>
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<h2>Number Line</h2>
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<p>With the <a href="../number-line.html">Number Line</a> we draw a thick line to show the values we are including, and:</p>
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<ul>
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<li>a filled-in circle when we want to include the end value, or</li>
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<li>an open circle when we don't</li>
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</ul>
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<p>Like this:</p>
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<div class="example">
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<p>Example:</p>
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<p align="center"><img src="images/interval-0-20.gif" alt="(0, 20)" width="249" height="54" /></p>
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<p>means all the numbers between 0 and 20, do <b>not</b> include 0, but <b>do</b> include 20</p>
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</div>
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<p> </p>
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<h2>All Three Methods Together</h2>
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<p>Here is a handy table showing all 3 methods (the interval is 1 to 2):</p>
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<div class="simple">
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<table width="100%" border="0" align="center">
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<tr>
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<td bgcolor="#FFFFFF"> </td>
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<td colspan="2" align="center" class="large">From 1</td>
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<td width="30" class="large"> </td>
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<td colspan="2" align="center" class="large">To 2</td>
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</tr>
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<tr>
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<td bgcolor="#FFFFFF"> </td>
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<td align="center">Including 1</td>
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<td align="center"><b>Not</b> Including 1</td>
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<td width="30"> </td>
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<td align="center"><b>Not</b> Including 2</td>
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<td align="center"> Including 2</td>
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</tr>
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<tr>
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<td class="large">Inequality:</td>
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<td align="center"><span class="large">x ≥ 1</span>
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<br /> "greater than
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<br /> or equal to"</td>
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<td align="center"><span class="large">x > 1</span>
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<br /> "greater than"
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<br /> </td>
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<td width="30" align="center"> </td>
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<td align="center"><span class="large">x < 2</span>
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<br /> "less than"
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<br /> </td>
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<td align="center"><span class="large">x ≤ 2</span>
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<br /> "less than
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<br /> or equal to"</td>
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</tr>
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<tr>
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<td class="large">Number line:</td>
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<td align="center"><img src="images/interval-ge-1.gif" alt="≥1" width="71" height="55" /></td>
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<td align="center"><img src="images/interval-gt-1.gif" alt=">1" width="70" height="55" /></td>
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<td width="30" align="center"> </td>
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<td align="center"><img src="images/interval-lt-2.gif" alt="<2" width="70" height="55" /></td>
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<td align="center"><img src="images/interval-le-2.gif" alt="≤2" width="70" height="55" /></td>
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</tr>
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<tr>
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<td class="large">Interval notation:</td>
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<td align="center" class="large">[1</td>
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<td align="center" class="large">(1</td>
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<td width="30" align="center" class="large"> </td>
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<td align="center" class="large">2)</td>
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<td align="center" class="large">2]</td>
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</tr>
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</table>
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</div>
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<p> </p>
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<p>Example: to <b>include 1</b>, and <b>not include 2</b>:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td class="large">Inequality:</td>
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<td align="center">
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<p><span class="large">x ≥ 1</span> and <span class="large">x < 2</span></p>
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<p>or together: <span class="large">1 </span><span class="large">≤</span><span class="large"> x < 2</span>
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<br /> </p>
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</td>
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</tr>
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<tr>
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<td class="large">Number line:</td>
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<td align="center"><img src="images/interval-ge-1.gif" alt="≥1" width="70" height="55" /><img src="images/interval-lt-2.gif" alt="<2" width="70" height="55" /></td>
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</tr>
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<tr>
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<td class="large">Interval notation:</td>
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<td align="center" class="large">[1, 2)</td>
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</tr>
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</table>
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</div>
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<h2>More Examples</h2>
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<div class="example">
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<h3>Example 1: "The Nothing Over $10 Sale"<br />
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</h3>
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<p>That means <b>up to and including</b> $10.</p>
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<p>And it is fair to say all prices are more than $0.00.</p>
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<p>As an <b>inequality</b> we show this as:</p>
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<p align="center" class="larger">Price ≤ 10 and Price > 0</p>
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<p align="center">In fact we could combine that into: </p>
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<p align="center" class="larger">0 < Price ≤ 10</p>
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<p>On the <b>number line</b> it looks like this:</p>
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<p align="center"><img src="images/interval-0-10.gif" alt="(0, 10]" width="242" height="52" /></p>
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<p>And using <b>interval notation</b> it is simply: </p>
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<p align="center"><span class="larger">(0, 10]</span></p>
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<p> </p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example 2: "Competitors must be between 14 and 18"</h3>
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<p>So 14 is included, and "being 18" goes all the way up to (but not including) 19.</p>
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<p>As an inequality it looks like this: </p>
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<p align="center" class="larger">14 ≤ Age < 19</p>
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<p>On the number line it looks like this:</p>
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<p align="center"><img src="images/interval-14-19.gif" alt="[14, 19)" width="247" height="51" /></p>
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<p>And using interval notation it is simply: </p>
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<p align="center" class="larger">[14, 19)</p>
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</div>
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<p><i>Isn't it funny how we measure age quite differently from anything else? We stay 18 right up until the moment we are fully 19. We don't we say we are 19 (to the nearest year) from 18½ onwards</i>. </p>
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<h2>Open or Closed</h2>
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<p>The terms "Open" and "Closed" are sometimes used when the end value is included or not:</p>
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<table border="0" align="center">
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<tr>
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<td align="center" class="larger">(a, b)</td>
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<td width="30" align="center" class="larger"> </td>
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<td align="center" class="larger">a < x < b</td>
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<td width="30"> </td>
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<td>an open interval</td>
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</tr>
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<tr>
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<td align="center" class="larger">[a, b)</td>
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<td align="center" class="larger"> </td>
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<td align="center" class="larger">a ≤ x < b</td>
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<td> </td>
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<td>closed on left, open on right</td>
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</tr>
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<tr>
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<td align="center" class="larger"> (a, b]</td>
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<td align="center" class="larger"> </td>
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<td align="center" class="larger">a < x ≤ b</td>
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<td> </td>
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<td>open on left, closed on right</td>
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</tr>
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<tr>
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<td align="center" class="larger"> [a, b]</td>
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<td align="center" class="larger"> </td>
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<td align="center" class="larger">a ≤ x ≤ b</td>
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<td> </td>
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<td>a closed interval </td>
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</tr>
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</table>
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<p>These are intervals of finite length. We also have intervals of infinite length. </p>
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<h2>To Infinity (but not beyond!) </h2>
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<p>We often use <a href="../numbers/infinity.html">Infinity</a> in interval notation. </p>
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<p>Infinity is <b>not a real number</b>, in this case it just means "continuing on ..." </p>
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<div class="example">
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<h3>Example: x greater than, or equal to, 3:</h3>
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<p align="center"> <span class="large">[3, +∞)</span></p>
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<p align="center"><img src="images/interval-3-inf.gif" alt="[3, infinity)" width="131" height="52" /></p>
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</div>
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<p>Note that we use the round bracket with infinity, because we don't reach it!</p>
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<p>There are 4 possible "infinite ends":</p>
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<table border="0" align="center">
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<tr>
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<td align="center" class="large">Interval</td>
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<td width="30" align="center" class="large"> </td>
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<td align="center" class="large">Inequality</td>
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<td width="30" align="center" class="large"> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center" class="large">(a, +∞)</td>
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<td align="center" class="large"> </td>
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<td align="center" class="large">x > a</td>
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<td align="center" class="large"> </td>
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<td align="center">"greater than a"</td>
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</tr>
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<tr>
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<td align="center" class="large">[a, +∞)</td>
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<td align="center" class="large"> </td>
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<td align="center" class="large">x ≥ a</td>
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<td align="center" class="large"> </td>
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<td align="center">"greater than or equal to a"</td>
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</tr>
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<tr>
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<td align="center" class="large">(-∞, a)</td>
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<td align="center" class="large"> </td>
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<td align="center" class="large">x < a</td>
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<td align="center" class="large"> </td>
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<td align="center">"less than a"</td>
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</tr>
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<tr>
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<td align="center" class="large">(-∞, a]</td>
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<td align="center" class="large"> </td>
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<td align="center" class="large">x ≤ a</td>
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<td align="center" class="large"> </td>
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<td align="center">"less than or equal to a"</td>
|
||
</tr>
|
||
</table>
|
||
<p>We could even show <b>no limits</b> by using this notation: <span class="large">(-∞, +∞)</span></p>
|
||
<h2>Two Intervals</h2>
|
||
<p>We can have two (or more) intervals.</p>
|
||
<div class="example">
|
||
<h3>Example: x ≤ 2 or x >3</h3>
|
||
<p>On the number line it looks like this:</p>
|
||
<p align="center"><img src="images/interval-to-2-3-up.gif" alt="two intervals" width="240" height="50" /></p>
|
||
<p>And interval notation looks like this:</p>
|
||
<p align="center" class="large"> (-∞, 2] U (3, +∞)</p>
|
||
<p>We used a "U" to mean Union (the joining together of two <a href="sets-introduction.html">sets</a>).</p>
|
||
</div>
|
||
<br />
|
||
<table border="0" align="center">
|
||
<tr>
|
||
<td>
|
||
<p><i>Note: be careful with inequalities like that one. <br>
|
||
Don't try to join it into one inequality: </i></p>
|
||
<p align="center" class="large"> 2 ≥ x > 3 <img src="../images/style/no.svg" alt="Not" /> wrong!</p>
|
||
<p><i>that doesn't make sense (you can't be less than 2 <br>
|
||
and greater than 3 at the same time).</i></p>
|
||
</td>
|
||
</tr>
|
||
</table>
|
||
<h2>Union and Intersection</h2>
|
||
<p>We just saw how to join two sets using "Union" (and the symbol <b>∪</b>).</p>
|
||
<p>There is also "Intersection" which means "has to be in both". <i>Think "where do they overlap?".</i></p>
|
||
<p>The Intersection symbol is an upside down "U" like this: <b>∩</b> </p>
|
||
<div class="example">
|
||
<h3>Example: (-∞, 6] ∩ (1, ∞)</h3>
|
||
<p>The first interval goes up to (and including) 6</p>
|
||
<p>The second interval goes from (but not including) 1 onwards.</p>
|
||
<div>
|
||
<div> </div>
|
||
</div>
|
||
<p align="center"><img src="images/interval-intersection.svg" alt="interval intersection" /></p>
|
||
<p>The <b>Intersection</b> (or overlap) of those two sets goes from 1 to 6 (not including 1, including 6):</p>
|
||
<p align="center" class="large"> (1, 6] </p>
|
||
</div>
|
||
<p> </p>
|
||
<h2>Conclusion</h2>
|
||
<ul>
|
||
<div class="bigul">
|
||
<li>An Interval is all the numbers between two given numbers.</li>
|
||
<li>Showing if the beginning and end number are included is important</li>
|
||
<li>There are three main ways to show intervals: Inequalities, The Number Line and Interval Notation.</li>
|
||
</div>
|
||
</ul>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script type="text/javascript">
|
||
getQ(451, 454, 1073, 1074, 1075, 1076, 2316, 2317, 2318, 2319);
|
||
</script> </div>
|
||
<p> </p>
|
||
<div class="center80">
|
||
<h3>Footnote: Geometry, Algebra and Sets</h3>
|
||
<p>You may not have noticed this ... but we have actually been using:</p>
|
||
<ul>
|
||
<li><a href="../algebra/index.html">Algebra</a>, </li>
|
||
<li><a href="../geometry/index.html">Geometry</a> (the Number Line is a <a href="../geometry/line.html">line</a>), and </li>
|
||
<li><a href="sets-introduction.html">Sets</a> (an interval is a set of numbers)</li>
|
||
</ul>
|
||
<p>all in one subject. Isn't mathematics amazing?</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="related"><a href="set-builder-notation.html">Set-Builder Notation</a> <a href="../algebra/index.html">Algebra Index</a> <a href="../algebra/inequality.html">Inequalities</a></div>
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