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 ∅.

Explanation:

Step 1: Divide both sides by -5

When dividing an inequality by a negative number, the inequality sign flips. So we have:
$$\frac{-5|5 - x|}{-5}>\frac{-10}{-5}$$
Simplifying gives:
$$|5 - x| > 2$$

Step 2: Rewrite the 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 get two inequalities:

1. \(5 - x < - 2\)
2. \(5 - x > 2\)

Step 3: Solve \(5 - x < - 2\)

Subtract 5 from both sides:
\(-x < - 2 - 5\)
\(-x < - 7\)
Multiply both sides by -1 (and flip the inequality sign):
\(x > 7\)

Step 4: Solve \(5 - x > 2\)

Subtract 5 from both sides:
\(-x > 2 - 5\)
\(-x > - 3\)
Multiply both sides by -1 (and flip the inequality sign):
\(x < 3\)

Answer:

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