Example

Solve for $x$: $\quad \displaystyle \frac{|3x+2|}{4} \le 1$

Solution

To solve an absolute value inequality like this one, we should first isolate the absolute value expression. We can do so by multiplying both sides of the inequality by $4$. Recall that multiplying by a positive does not reverse the inequality.

This inequality tells us that the number between the absolute value bars is no more than 4 units from $0$. In other words,

This compound inequality is an "AND" inequality, and it can be solved by subtracting $2$ and then dividing by $3$ all the way across the inequalities.

In interval notation, this solution is $[-2,2/3]$, and the corresponding number line is