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      <text>3.5.1.1 The Inequality Symbols</text>
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      <text><![CDATA[<h4>The inequality symbols</h4>
<br>
<p>Recall the meaning of the following symbols:</p>
<br>
<div class="HELM_keypoint">
    <h4>Key Point</h4>
    <p>The symbols <span class="math inline">\(&gt;\)</span>, <span class="math inline">\(&lt;\)</span>, <span class="math inline">\(\ge\)</span>, <span class="math inline">\(\le\)</span> are called <span><strong>inequalities</strong></span></p>
    <p><span class="math inline">\(&gt;\)</span> means: `is greater than'</p>
    <p><span class="math inline">\(≥\)</span> means: `is greater than or equal to'</p>
    <p><span class="math inline">\(&lt;\)</span> means: `is less than'</p>
    <p><span class="math inline">\(≤\)</span> means: `is less than or equal to'</p>
</div>
<p>So, we may state, for example, <span class="math display">\[8&gt;7\qquad 9\ge 2\qquad -2&lt;3\qquad 
7\le 7\]</span></p>
<p>A number line is often a helpful way of picturing inequalities. Given two numbers <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>, if <span class="math inline">\(b&gt;a\)</span> then <span class="math inline">\(b\)</span>    will be to the right of <span class="math inline">\(a\)</span> on the number line as shown in the figure below.&nbsp;</p>
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    <img src="@@PLUGINFILE@@/3.5_fig1%20%282%29.svg" alt="" width="331" height="20" role="presentation" class="img-responsive atto_image_button_text-bottom">
</figure>
<p>Note from the figure below that −3 &gt; −5, 4 &gt; −2 and 8 &gt; 5.</p>
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    <p><img src="@@PLUGINFILE@@/3.5_fig2%20%281%29.svg" alt="" width="493" height="28" role="presentation" class="img-responsive atto_image_button_text-bottom"><br></p>
</figure>
<p>Inequalities can always be written in two ways. For example in English we can state that 8 is greater than 7, or equivalently, that 7 is less than 8. Mathematically we write <span class="math inline">\(8&gt;7\)</span> or <span class="math inline">\(7&lt;8\)</span>.
    Similarly if <span class="math inline">\(b&gt;a\)</span> then <span class="math inline">\(a&lt;b\)</span>. If <span class="math inline">\(a&lt;b\)</span> then <span class="math inline">\(a\)</span> will be to the left of <span class="math inline">\(b\)</span>    on the number line.</p>
<hr>
<h4 class="HELM_example">Example</h4>
<p>Rewrite the inequality <span class="math inline">\(-{2\over 5}&lt;x\)</span> using only the ‘greater than’ sign, <span class="math inline">\(&gt;\)</span>.</p>
<h4 class="HELM_solution">Solution</h4>
<p><span class="math inline">\(-{2\over 5}&lt;x\)</span> can be written as <span class="math inline">\(x&gt;-{2\over 5}\)</span></p>

<hr>
<h4 class="HELM_example">Example</h4>
<p>Rewrite the inequality <span class="math inline">\(5&gt;x\)</span> using only the ‘less than’ sign, <span class="math inline">\(&lt;\)</span>.</p>
<h4 class="HELM_solution">Solution</h4>
<p><span class="math inline">\(5&gt;x\)</span> can be written as <span class="math inline">\(x&lt;5\)</span>.</p>

<hr>
<p>Sometimes two inequalities are combined into a single statement. Consider for example the statement <span class="math inline">\(3~&lt;~x~&lt;~6\)</span>. This is a compact way of writing ‘<span class="math inline">\(3&lt;x\)</span> and <span class="math inline">\(x&lt;6\)</span>’.
    Now <span class="math inline">\(3&lt;x\)</span> is equivalent to <span class="math inline">\(x&gt;3\)</span> and so <span class="math inline">\(3&lt;x&lt;6\)</span> means <span class="math inline">\(x\)</span> is greater than 3 but less than 6.</p>
<br>
<p>Inequalities obey simple rules when used in conjunction with arithmetical operations:</p>
<div class="HELM_keypoint">
    <h4>Key Point</h4>
    <ol>
        <li>
            <p>Adding or subtracting the same quantity from both sides of an inequality leaves the inequality sign unchanged.</p>
        </li>
        <li>
            <p>Multiplying or dividing both sides by a <span><strong>positive</strong></span> number leaves the inequality sign unchanged.</p>
        </li>
        <li>
            <p>Multiplying or dividing both sides by a <span><strong>negative</strong></span> number reverses the inequality.</p>
        </li>
    </ol>
</div>

<p>For example, since <span class="math inline">\(8&gt;5\)</span>, by adding <span class="math inline">\(k\)</span> to both sides we can state <span class="math display">\[8 + k &gt; 5 + k\]</span> for any value of <span class="math inline">\(k\)</span>.
    For example (with <span class="math inline">\(k=-3\)</span>) <span class="math inline">\(8-3&gt;5-3\)</span>.</p>
<p>Further, by multiplying both sides of <span class="math inline">\(8&gt;5\)</span> by <span class="math inline">\(k\)</span> we can state <span class="math display">\[8k &gt; 5k\]</span> provided <span class="math inline">\(k\)</span> is positive.</p>
<p>However,&nbsp;<span class="math display">\[8k&lt;5k\]</span> if <span class="math inline">\(k\)</span> is negative. We emphasise that the inequality sign is reversed when multiplying both sides by a negative number. A common mistake is to forget to reverse
    the inequality sign when multiplying inequalities by a negative number. For example <span class="math inline">\(8&gt;5\)</span>, but multiplying both sides by <span class="math inline">\(-1\)</span> gives <span class="math inline">\(-8&lt;-5\)</span>.</p>]]></text>
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