Question

solve the absolute value inequality
$-5|5 - x| < -10$

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 $varnothing$

Explanation:

Step1: Divide both sides by -5 (reverse inequality)

When we divide both sides of an inequality by a negative number, the direction of the inequality sign reverses. So, dividing both sides of \(-5|5 - x| < -10\) by \(-5\), we get \(|5 - x| > 2\).

Step2: Rewrite absolute value inequality

The absolute value inequality \(|A| > B\) (where \(B>0\)) is equivalent to \(A > B\) or \(A < -B\). So, for \(|5 - x| > 2\), we have two cases:

• Case 1: \(5 - x > 2\)

Subtract 5 from both sides: \(-x > 2 - 5\) which simplifies to \(-x > -3\). Multiply both sides by -1 (and reverse the inequality sign): \(x < 3\).

• Case 2: \(5 - x < -2\)

Subtract 5 from both sides: \(-x < -2 - 5\) which simplifies to \(-x < -7\). Multiply both sides by -1 (and reverse the inequality sign): \(x > 7\).

Answer:

A. The solution set in interval notation is \((-\infty, 3) \cup (7, \infty)\)