<p class="paragraph-header">Before clearing denominators</p>
<p>A fraction is undefined when its denominator is zero. State these excluded values before multiplying an equation by a denominator.</p>
<p>For&nbsp;\(\dfrac{3}{x-1}+\dfrac{2}{x+2}=1\), the restrictions are \(x\ne1\) and \(x\ne-2\). Multiplication by \((x-1)(x+2)\) is valid only for the allowed values.</p>
<p>After solving the resulting equation, reject any root excluded by the original denominators. Cancellation does not restore an excluded value.</p>
<p class="paragraph-header">From a situation to an equation</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>Step</strong></th><th style="text-align: center; --cell-justify: center"><strong>What to record</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Define</td><td style="text-align: center; --cell-justify: center">State what the unknown measures and its unit.</td></tr><tr><td style="text-align: center; --cell-justify: center">Restrict</td><td style="text-align: center; --cell-justify: center">Use the situation: lengths, times and prices must be positive; a width may also have an upper limit.</td></tr><tr><td style="text-align: center; --cell-justify: center">Form</td><td style="text-align: center; --cell-justify: center">Express related quantities in the same units, then write an equation.</td></tr><tr><td style="text-align: center; --cell-justify: center">Interpret</td><td style="text-align: center; --cell-justify: center">Solve, reject impossible roots, and answer the original question with units.</td></tr></tbody></table></div></div>
<p>A positive root can still be impossible. For example, a border inside a rectangle must be narrower than half its shorter side.</p>
<p>To compare journey times, use \(\text{time}=\dfrac{\text{distance}}{\text{speed}}\). Convert minutes to hours before combining terms measured in hours.</p>

A fraction is undefined when its denominator is zero. State these excluded values before multiplying an equation by a denominator.

For \(\dfrac{3}{x-1}+\dfrac{2}{x+2}=1\), the restrictions are \(x\ne1\) and \(x\ne-2\). Multiplication by \((x-1)(x+2)\) is valid only for the allowed values.

After solving the resulting equation, reject any root excluded by the original denominators. Cancellation does not restore an excluded value.

A positive root can still be impossible. For example, a border inside a rectangle must be narrower than half its shorter side.

To compare journey times, use \(\text{time}=\dfrac{\text{distance}}{\text{speed}}\). Convert minutes to hours before combining terms measured in hours.

Restrictions and mathematical models

Restrictions and mathematical models

Step	What to record
Define	State what the unknown measures and its unit.
Restrict	Use the situation: lengths, times and prices must be positive; a width may also have an upper limit.
Form	Express related quantities in the same units, then write an equation.
Interpret	Solve, reject impossible roots, and answer the original question with units.