Question

solve the following absolute value inequality.
$3|x - 7| \leq 27$
$x \leq \boldsymbol{?}$ and $x \geq \boldsymbol{?}$

Explanation:

Step1: Isolate absolute value term

Divide both sides by 3:
$\frac{3|x-7|}{3} \leq \frac{27}{3}$
Simplify to get:
$|x-7| \leq 9$

Step2: Rewrite without absolute value

For $|A| \leq B$, this means $-B \leq A \leq B$:
$-9 \leq x-7 \leq 9$

Step3: Solve upper bound

Add 7 to the right inequality:
$x-7 \leq 9$
$x \leq 9 + 7$
$x \leq 16$

Step4: Solve lower bound

Add 7 to the left inequality:
$x-7 \geq -9$
$x \geq -9 + 7$
$x \geq -2$

Answer:

$x \leq 16$ and $x \geq -2$