audio read-through Adding and Subtracting Fractions With Variables

To add or subtract fractions:


Question: Combine into a single fraction: $$ \cssId{s12}{\frac{2}{x+3} - \frac{3x}{x-1}} $$
Solution: Note that the LCD is $\,(x+3)(x-1)\,.$
$\displaystyle\frac{2}{x+3} - \frac{3x}{x-1}$ original expression
$\displaystyle = \frac{2}{x+3}\cdot\frac{x-1}{x-1} $

$\displaystyle \qquad -\ \frac{3x}{x-1}\cdot\frac{x+3}{x+3} $
get a common denominator by multiplying by $\,1\,$
$\displaystyle = \frac{2(x-1)-3x(x+3)}{(x+3)(x-1)}$ keep the denominator the same; add the numerators
$\displaystyle = \frac{2x-2-3x^2 - 9x}{(x+3)(x-1)}$ multiply out the numerator
$\displaystyle = \frac{-3x^2 - 7x - 2}{(x+3)(x-1)}$ combine like terms; write numerator in standard form

Leave the denominator in factored form for your final answer.

Concept Practice