If |2x - 3| ≤ 9, then which of the following intervals represents all possible values of the expression 5x + 7?

Question: If |2x - 3| ≤ 9, then which of the following intervals represents all possible values of the expression 5x + 7?

Solution:
Start with the given inequality:
|2x - 3| ≤ 9

Rewrite as a compound inequality:
- 9 ≤ 2x - 3 ≤ 9
⇒ - 9 + 3 ≤ 2x - 3 + 3 ≤ 9 + 3
⇒ - 6 ≤ 2x ≤ 12
⇒ - 3 ≤ x ≤ 6

Now, find the range of 5x + 7:
Multiply the interval by 5:
5(- 3) ≤ 5x ≤ 5(6)
⇒ - 15 ≤ 5x ≤ 30
⇒ - 15 + 7 ≤ 5x + 7 ≤ 30 + 7
⇒ - 8 ≤ 5x + 7 ≤ 37

So, all possible values of 5x + 7 lie in the interval: [ - 8, 37 ]