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<h1 class="center">Solving Inequality Word Questions</h1>
<p class="center"><i>(You might like to read <a href="inequality.html">Introduction to Inequalities</a> and <a href="inequality-solving.html">Solving Inequalities</a> first.)</i></p><br>
<p>In Algebra we have "inequality" questions like:</p>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="../images/soccer-teams.jpg" alt="soccer teams" height="86" width="209"></p>
<h3>Sam and Alex play in the same soccer team. <br>
Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. <br>
<br>
What are the possible number of goals Alex scored?</h3>
</div>
<p>How do we solve them?</p>
<p>The trick is to break the solution into two parts:</p>
<div class="center80">
<p class="center large">Turn the English into Algebra.</p>
<p class="center large">Then use Algebra to solve.</p>
</div>
<h2>Turning English into Algebra</h2>
<p>To turn the English into Algebra it helps to:</p>
<ul>
<li>Read the whole thing first</li>
<li>Do a sketch if needed</li>
<li>Assign <b>letters</b> for the values</li>
<li>Find or work out <b>formulas</b></li>
</ul>
<p>We should also write down <b>what is actually being asked for</b>, so we know where we are going and when we have arrived!</p>
<p>&nbsp;</p>
<p>The best way to learn this is by example, so let's try our first example:</p>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="../images/soccer-teams.jpg" alt="soccer teams" height="86" width="209"></p>
<h3>Sam and Alex play in the same soccer team. <br>
Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. <br>
<br>
What are the possible number of goals Alex scored?</h3>
<p>&nbsp;</p>
<p>Assign Letters:</p>
<ul>
<li>the number of goals Alex scored: <b>A</b></li>
<li>the number of goals Sam scored: <b>S</b></li>
</ul>
<p>We know that Alex scored 3 more goals than Sam did, so: <b>A = S + 3</b></p>
<p>And we know that together they scored less than 9 goals: <b>S + A &lt; 9</b></p>
<p>We are being asked for how many goals Alex might have scored: <b>A</b></p>
<p>&nbsp;</p>
<h4>Solve:</h4>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right"><b>S + A &lt; 9</b></span></div>
<div class="row"><span class="left">A = S + 3, so:</span><span class="right"><b>S + <span class="hilite">(S + 3)</span> &lt; 9</b></span></div>
<div class="row"><span class="left">Simplify:</span><span class="right"><b>2S + 3 &lt; 9</b></span></div>
<div class="row"><span class="left">Subtract 3 from both sides:</span><span class="right"><b>2S &lt; 9 &minus; 3</b></span></div>
<div class="row"><span class="left">Simplify:</span><span class="right"><b>2S &lt; 6</b></span></div>
<div class="row"><span class="left">Divide both sides by 2:</span><span class="right"><b>S &lt; 3</b></span></div>
</div>
<p>Sam scored less than 3 goals, which means that Sam could have scored 0, 1 or 2 goals.</p>
<p>Alex scored 3 more goals than Sam did, so <b>Alex could have scored 3, 4, or 5 goals</b>.</p>
<p>&nbsp;</p>
<p>Check:</p>
<ul>
<li>When S = 0, then <b>A = 3</b> and S + A = 3, and 3 &lt; 9 is correct</li>
<li>When S = 1, then <b>A = 4</b> and S + A = 5, and 5 &lt; 9 is correct</li>
<li>When S = 2, then <b>A = 5</b> and S + A = 7, and 7 &lt; 9 is correct</li>
<li>(But when S = 3, then A = 6 and S + A = 9, and 9 &lt; 9 is incorrect)</li>
</ul>
</div>
<h2>Lots More Examples!</h2>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/pups.jpg" alt="pups" height="125" width="239"></p>
<h3>Example: Of 8 pups, there are more girls than boys.<br>
<br>
How many girl pups could there be?</h3>
<p>Assign Letters:</p>
<ul>
<li>the number of girls: <b>g </b></li>
<li>the number of boys: <b>b</b></li>
</ul>
<p>We know that there are 8 pups, so: g + b = 8, which can be rearranged to</p>
<div class="so">b = 8 &minus; g</div>
<p>We also know there are more girls than boys, so:</p>
<div class="so"> g &gt; b</div>
<p>We are being asked for the number of girl pups: <b>g</b></p>
<p>Solve:</p>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right"><b>g &gt; b</b></span></div>
<div class="row"><span class="left"><b>b = 8 &minus; g</b>, so:</span><span class="right"><b>g &gt; <span class="hilite">8 &minus; g</span></b></span></div>
<div class="row"><span class="left">Add g to both sides:</span><span class="right"><b>g + g &gt; 8</b></span></div>
<div class="row"><span class="left">Simplify:</span><span class="right"><b>2g &gt; 8</b></span></div>
<div class="row"><span class="left">Divide both sides by 2:</span><span class="right"><b>g &gt; 4</b></span></div>
</div>
<p>So there could be 5, 6, 7 or 8 girl pups.</p>
<p>Could there be 8 girl pups? Then there would be no boys at all, and the question isn't clear on that point&nbsp;(sometimes questions are like that).</p>
<p>Check</p>
<ul>
<li>When g = 8, then b = 0 and g &gt; b is correct (but is b = 0 allowed?)</li>
<li>When g = 7, then b = 1 and g &gt; b is correct</li>
<li>When g = 6, then b = 2 and g &gt; b is correct</li>
<li>When g = 5, then b = 3 and g &gt; b is correct</li>
<li>(But if g = 4, then b = 4 and g &gt; b is incorrect)</li>
</ul>
</div>
<p>A speedy example:</p>
<div class="example">
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/bike.jpg" alt="bike" height="100" width="154"></p>
<h3>Example: Joe enters a race where he has to cycle and run.<br>
<br>
He cycles a distance of 25 km, and then runs for 20 km. His average running speed is half of his average cycling speed.<br>
<br>
Joe completes the race in less than 2&frac12; hours, what can we say about his average speeds?</h3>
<p>Assign Letters:</p>
<ul>
<li>Average running speed: <b>s</b></li>
<li>So average cycling speed: <b>2s</b></li>
</ul>
<p>Formulas:</p>
<ul>
<li>Speed = <span class="intbl"><em>Distance</em><strong>Time</strong></span></li>
<li>Which can be rearranged to: Time = <span class="intbl"><em>Distance</em><strong>Speed</strong></span></li>
</ul>
<p>We are being asked for his average speeds: <b>s</b> and <b>2s</b></p>
<p>&nbsp;</p>
<p>The race is divided into two parts:</p>
<h4>1. Cycling</h4>
<ul>
<li>Distance = 25 km</li>
<li>Average speed = 2s km/h</li>
<li>So Time = <span class="intbl"><em>Distance</em><strong>Average Speed</strong></span> = <span class="intbl"><em>25</em><strong>2s</strong></span> hours</li>
</ul>
<h4>2. Running</h4>
<ul>
<li>Distance = 20 km</li>
<li>Average speed = s km/h</li>
<li>So Time = <span class="intbl"><em>Distance</em><strong>Average Speed</strong></span> = <span class="intbl"><em>20</em><strong>s</strong></span> hours</li>
</ul>
<p>Joe completes the race in less than 2&frac12; hours</p>
<ul>
<li>The total time &lt; 2&frac12;</li>
<li><span class="intbl"><em>25</em><strong>2s</strong></span> + <span class="intbl"><em>20</em><strong>s</strong></span> &lt; 2&frac12;</li>
</ul>
<p>Solve: </p>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right"><b><span class="intbl"><em>25</em><strong>2s</strong></span> + <span class="intbl"><em>20</em><strong>s</strong></span> &lt; 2&frac12;</b></span></div>
<div class="row"><span class="left">Multiply all terms by 2s:</span><span class="right"><b>25 + 40 &lt; 5s </b></span></div>
<div class="row"><span class="left">Simplify:</span><span class="right"><b>65 &lt; 5s</b></span></div>
<div class="row"><span class="left">Divide both sides by 5:</span><span class="right"><b>13 &lt; s</b></span></div>
<div class="row"><span class="left">Swap sides:</span><span class="right"><b>s &gt; 13</b></span></div>
</div>
<p>So his average speed running is greater than 13 km/h and his average speed cycling is greater than 26 km/h</p>
</div>
<p>In this example we get to use two inequalities at once:</p>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/ball-throw.jpg" alt="ball throw" height="283" width="110"></p>
<h3>Example: The velocity <b>v</b> m/s of a ball thrown directly up in the air is given by <b>v = 20 &minus; 10t</b>, where <b>t</b> is the time in seconds.<br>
<br>
At what times will the velocity be between 10 m/s and 15 m/s?</h3>
<p>Letters:</p>
<ul>
<li>velocity in m/s: <b>v</b></li>
<li>the time in seconds: <b>t</b></li>
</ul>
<p>Formula:</p>
<ul>
<li><b>v = 20 &minus; 10t</b></li>
</ul>
<p>We are being asked for the time <b>t</b> when <b>v</b> is between 5 and 15 m/s:</p>
<div class="so">10 &nbsp;&lt;&nbsp; v &nbsp;&lt;&nbsp; 15</div>
<div class="so">10 &nbsp;&lt;&nbsp; 20 &minus; 10t &nbsp;&lt;&nbsp; 15</div>
<p>Solve:</p>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right"><b>10 &nbsp;&lt;&nbsp; 20 &minus; 10t &nbsp;&lt;&nbsp; 15</b></span></div>
<div class="row"><span class="left">Subtract 20 from each:</span><span class="right"><b>10 <span class="hilite">&minus; 20</span> &nbsp;&lt;&nbsp; 20 &minus; 10t <span class="hilite">&minus; 20</span> &nbsp;&lt;&nbsp; 15 <span class="hilite">&minus; 20</span></b></span></div>
<div class="row"><span class="left">Simplify:</span><span class="right"><b>&minus;10 &nbsp;&lt;&nbsp;&minus;10t &nbsp;&lt;&nbsp; &minus;5</b></span></div>
<div class="row"><span class="left">Divide each by 10:</span><span class="right"><b>&minus;1 &nbsp;&lt;&nbsp;&minus;t &nbsp;&lt;&nbsp; &minus;0.5</b></span></div>
<div class="row"><span class="left">Change signs and reverse inequalities:</span><span class="right"><b>1 &nbsp;<span class="hilite">&gt;</span>&nbsp; t &nbsp;<span class="hilite">&gt;</span>&nbsp; 0.5</b></span></div>
<div class="row"><span class="left"><span class="center">It is neater to show the smaller<br>
number first,
so</span> swap over:</span><span class="right"><b>0.5 &nbsp;&lt;&nbsp; t &nbsp;&lt;&nbsp; 1</b></span></div>
</div>
<p>So the velocity is between 10 m/s and 15 m/s between 0.5 and 1 second after.</p>
</div>
<p>And a reasonably <b>hard</b> example to finish with:</p>
<div class="example">
<h3>Example: A rectangular room fits at least 7 tables that each have 1 square meter of surface area.
The perimeter of the room is 16 m.<br>
What could the width and length of the room be?</h3>
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/room-size.gif" alt="room size" height="107" width="158"></p>
<p>Make a sketch: we don't know the size of the tables, only their area, they may fit perfectly or not!</p>
<p>Assign Letters:</p>
<ul>
<li>the length of the room: <b>L</b></li>
<li>the width of the room: <b>W</b></li>
</ul>
<p>The formula for the perimeter is <b>2(W + L)</b>, and we know it is 16 m</p>
<ul>
<li>2(W + L) = 16</li>
<li>W + L = 8</li>
<li>L = 8 &minus; W</li>
</ul>
<p>We also know the area of a rectangle is the width times the length: <b>Area = W &times; L </b></p>
<p>And the area must be greater than or equal to 7:</p>
<ul>
<li>W &times; L &ge; 7</li>
</ul>
<p>We are being asked for the possible values of <b>W</b> and <b>L</b></p>
<p>&nbsp;</p>
<p>Let's solve:</p>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right"><b>W &times; L &ge; 7</b></span></div>
<div class="row"><span class="left">Substitute L = 8 &minus; W:</span><span class="right"><b>W &times; <span class="hilite">(8 &minus; W)</span> &ge; 7</b></span></div>
<div class="row"><span class="left">Expand:</span><span class="right"><b>8W &minus; W<sup>2</sup> &ge; 7</b></span></div>
<div class="row"><span class="left">Bring all terms to left hand side:</span><span class="right"><b>W<sup>2</sup> &minus; 8W + 7 &le; 0</b></span></div>
</div>
<p>This is a quadratic inequality. It can be solved many way, here we will solve it by <a href="completing-square.html">completing the square</a>:</p>
<div class="tbl">
<div class="row"><span class="left">Move the number term <b>&minus;</b>7 to the right side of the inequality:</span><span class="right"><b>W<sup>2</sup> &minus; 8W &le; &minus;7</b></span></div>
<div class="row"><span class="left">Complete the square on the left side of the inequality and balance this by adding the same value to the
right side of the inequality:</span><span class="right"><b>W<sup>2</sup> &minus; 8W + 16 &le; &minus;7 + 16</b></span></div>
<div class="row"><span class="left">Simplify:</span><span class="right"><b>(W &minus; 4)<sup>2</sup> &le; 9</b></span></div>
<div class="row"><span class="left">Take the square root on both sides of the inequality:</span><span class="right"><b>&minus;3 &le; W &minus; 4 &le; 3</b></span></div>
<div class="row"><span class="left">
<p class="larger">Yes we have two inequalities, because <b>3<sup>2</sup> = 9</b> AND <b>(&minus;3)<sup>2</sup> = 9</b></p></span></div>
<div class="row"><span class="left">Add 4 to both sides of each inequality:</span><span class="right"><b>1 &le; W &le; 7</b></span></div>
</div>
<p>So the width must be <b>between 1 m and 7 m</b> (inclusive) and the length is <b>8&minus;width</b>.</p>
<p>&nbsp;</p>
<p>Check:</p>
<ul>
<li>Say W = 1, then L = 8&minus;1 = 7, and A = 1 x 7 = 7 m<sup>2</sup> (fits exactly 7 tables)</li>
<li>Say W = 0.9 (less than 1), then L = 7.1, and A = 0.9 x 7.1 = 6.39 m<sup>2</sup> (7 won't fit)</li>
<li>Say W = 1.1 (just above 1), then L = 6.9, and A = 1.1 x 6.9 = 7.59 m<sup>2</sup> (7 fit easily)</li>
<li>Likewise for W around 7 m</li>
</ul>
</div>
<p>&nbsp;</p>
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<a href="inequality.html">Inequalities</a>
<a href="inequality-solving.html">Solving Inequalities</a>
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<a href="../data/inequality-grapher.html">Inequality Grapher</a>
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