# What is the solution set for abs(3x – 2) < 7?

Aug 22, 2015

$x \in \left(- \frac{5}{3} , 3\right)$

#### Explanation:

Since you're dealing with the absolute value of a variable expression, you must take into account the fact that this expression can be negative or positive.

• $3 x - 2 > 0 \implies | 3 x - 2 | = 3 x - 2$

The inequality becomes

$3 x - 2 < 7$

$3 x < 9 \implies x < 3$

• $\left(3 x - 2\right) < 0 \implies | 3 x - 2 | = - \left(3 x - 2\right)$

This time, you get that

$- \left(3 x - 2\right) < 7$

$- 3 x + 2 < 7$

$- 3 x < 5 \implies x > - \frac{5}{3}$

This means that your original inequality will be true for any value of $x$ that is smaller than $3$ and bigger than $- \frac{5}{3}$.

The solution interval will thus be $\left(- \frac{5}{3} , 3\right)$.