Number line & inequalities

<p class="paragraph-header">Preserving the direction</p>
<p>Adding or subtracting the same number on both sides preserves an inequality. Multiplying or dividing by a positive number also preserves its direction.</p>
<p class="paragraph-header">Reversing the direction</p>
<p>Multiplying or dividing by a negative number reverses the comparison. For example, \(2&lt;5\), but \(-2&gt;-5\). On a number line, multiplication by \(-1\) reverses the order.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Operation</strong></th><th style="text-align: center; --cell-justify: center"><strong>Result</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(a&lt;b\); add \(c\)</td><td style="text-align: center; --cell-justify: center">\(a+c&lt;b+c\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(a&lt;b\); multiply by \(c&gt;0\)</td><td style="text-align: center; --cell-justify: center">\(ac&lt;bc\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(a&lt;b\); multiply by \(c&lt;0\)</td><td style="text-align: center; --cell-justify: center">\(ac&gt;bc\)</td></tr></tbody></table></div></div>
<p>Apply an operation to every expression in a double inequality. After division by a negative number, rewrite the result with the smaller boundary on the left. Multiplication by zero does not preserve an equivalent inequality; the sign of a multiplier must be known.</p>

Adding or subtracting the same number on both sides preserves an inequality. Multiplying or dividing by a positive number also preserves its direction.

Multiplying or dividing by a negative number reverses the comparison. For example, \(2<5\), but \(-2>-5\). On a number line, multiplication by \(-1\) reverses the order.

Apply an operation to every expression in a double inequality. After division by a negative number, rewrite the result with the smaller boundary on the left. Multiplication by zero does not preserve an equivalent inequality; the sign of a multiplier must be known.

Operation	Result
\(a<b\); add \(c\)	\(a+c<b+c\)
\(a<b\); multiply by \(c>0\)	\(ac<bc\)
\(a<b\); multiply by \(c<0\)	\(ac>bc\)

Equivalent inequalities

Equivalent inequalities