Question

solve the absolute value inequality
-6|7 - x| < -30

select the correct answer below and, if necessary, fill in the answer box to complete your choice
a. the solution set in interval notation is
(simplify your answer.)
b. the solution set is ∅

Explanation:

Step1: Divide both sides by -6, reverse inequality

$\frac{-6|7-x|}{-6} > \frac{-30}{-6}$
Simplify to: $|7-x| > 5$

Step2: Rewrite absolute value inequality

This splits into two inequalities:
$7-x > 5$ or $7-x < -5$

Step3: Solve first inequality

Subtract 7 from both sides: $-x > 5-7$
Simplify: $-x > -2$
Multiply by -1, reverse inequality: $x < 2$

Step4: Solve second inequality

Subtract 7 from both sides: $-x < -5-7$
Simplify: $-x < -12$
Multiply by -1, reverse inequality: $x > 12$

Step5: Write in interval notation

Combine solutions: $(-\infty, 2) \cup (12, \infty)$

Answer:

A. The solution set in interval notation is $(-\infty, 2) \cup (12, \infty)$