 # Equations with Fractions and Parentheses

Here you’ll learn how to solve equations with fractions. The only major difference from what you’ve done so far is to find a common denominator and multiply all the terms with it to be able to cancel out the denominators. The whole point of this method is to get rid of the fractions!

Rule

### SolvingEquationswithFractionsandParentheses

1.
Find the common denominator.
2.
Multiply all terms with the common denominator and simplify the denominators. If there’s anything left of the common denominator, you have to multiply the rest with the numerator. Notice how I put parentheses around the numerators.
3.
Think PEMDAS! Solve the parentheses first.
4.
Move all the terms with only numbers to one side. Remember to change signs.
5.
Move all the terms containing $x$ to the other side. Remember to change signs.
6.
Simplify both sides.
7.
Multiply or divide both sides with the number that prevents $x$ from standing by itself.

Example 1

Solve the equation $\frac{x}{3}+1=4-\frac{2x}{3}$

$\begin{array}{llll}\hfill \frac{x}{3}+1& =4-\frac{2x}{3}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 3×\phantom{\rule{-0.17em}{0ex}}\left(\frac{x}{3}+1\right)& =3×\phantom{\rule{-0.17em}{0ex}}\left(4-\frac{2x}{3}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{\text{3}x}{\text{3}}+3& =12-\frac{2x×\text{3}}{\text{3}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x+3& =12-2x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x+2x& =12-3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 3x& =9\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$