# Solve the absolute value of inequalities and express in interval notation for abs(3(x+2)-7x) <=6?

May 5, 2015

Simplify the expression within the absolute value then consider the two cases where that expression has a negative and a non-negative value.

$\left\mid 3 \left(x + 2\right) - 7 x \right\mid \le 6$

$\left\mid 6 - 4 x \right\mid \le 6$

Case 1
If $6 - 4 x < 0$
then $\left\mid 6 - 4 x \right\mid = 4 x - 6$
and $x > \frac{3}{2}$

$4 x - 6 \le 6$
$4 x \le 12$
$x \le 3$

So in this case $\frac{3}{2} < x \le 3$

Case 2
If $6 - 4 x \ge 0$
then $\left\mid 6 - 4 x \right\mid = 6 - 4 x$
and $x \le \frac{3}{2}$

$6 - 4 x \le 6$
$- 4 x \le 0$
$x \ge 0$

In this case $0 \le x \le \frac{3}{2}$

Combining the cases
Valid solutions are values of $x$ for which
either $\frac{3}{2} < x \le 3$
or $0 \le x \le \frac{3}{2}$

which combines as
$0 \le x \le 3$