Question

How do I do this? Solve each inequality. Write the solutions as either the union or intersection of two sets. (Show work)

|2x – 7| < 11
|5b + 8| ≥ 17
3|m – 4| ≤ 18
a) |x| < 9
b) |x| > 9
c) |x| < 11
d) |x| > 11

Answer 1

To solve absolute value inequalities, we consider two cases: when the expression inside the absolute value is positive or zero, and when it is negative. For each inequality, we apply these cases and find the solution sets.

Step-by-step explanation:

To solve the inequalities |2x – 7| < 11, |5b + 8| ≥ 17, and 3|m – 4| ≤ 18, we need to consider the two possible cases for absolute value inequalities:

Case 1: If the expression inside the absolute value is positive or zero, we can remove the absolute value symbols without changing the inequality. Case 2: If the expression inside the absolute value is negative, we need to multiply both sides of the inequality by -1 and flip the direction of the inequality sign.

Let's solve each inequality one by one:

a) For |x| < 9:

Case 1: When x is positive or zero, we have x < 9. Case 2: When x is negative, we have -x < 9, which simplifies to x > -9. Therefore, the solution set is (-9, 9).

b) For |x| > 9:

Case 1: When x is positive or zero, we have x > 9. Case 2: When x is negative, we have -x > 9, which simplifies to x < -9. Therefore, the solution set is (-∞, -9) ∪ (9, ∞).

c) For |x| < 11:

Case 1: When x is positive or zero, we have x < 11. Case 2: When x is negative, we have -x < 11, which simplifies to x > -11. Therefore, the solution set is (-11, 11).

d) For |x| > 11:

Case 1: When x is positive or zero, we have x > 11. Case 2: When x is negative, we have -x > 11, which simplifies to x < -11. Therefore, the solution set is (-∞, -11) ∪ (11, ∞).