Please explain these abs value inequalities?

Question

1.) |2x − 3| ≤ 0.2

2.) |3x − 2| ≥ 5

Answer 1

(1).

The absolute inequality is | 2x - 3 | ≤ 0.2.

| 2x - 3 | ≤ 0.2 is equivalent to - 0.2 ≤ 2x - 3 ≤ 0.2.

Solve both parts at the same time by adding 3 from each part. Then divide each part by 2.

- 0.2 + 3 ≤ 2x - 3 + 3 ≤ 0.2 + 3

2.8 ≤ 2x ≤ 3.2

2.8/2 ≤ 2x/2 ≤ 3.2/2

1.4 ≤ x ≤ 1.6

The solution set is {x | 1.4 ≤ x ≤ 1.6} and its graph is

Answer 2

(2).

The absolute inequality is | 3x - 2 | ≥ 5.

| 3x - 2 | ≥ 5 is equivalent to 3x - 2 ≥ 5 or 3x - 2 ≤ - 5.

Solve the inequality 1 : 3x - 2 ≥ 5 for x.

Add 2 to each side.

3x - 2 + 2 ≥ 5 + 2

3x ≥ 7

Divide each side by 3.

3x/3 ≥ 7/3

x ≥ 7/3 ≅ 2.33

Solve the inequality 2 : 3x - 2 ≤ - 5 for x.

Add 3 to each side.

3x - 2 + 2 ≤ - 5 + 2

3x ≤ - 3

Divide each side by 3.

3x/3 ≤ - 3/3

x ≤ 1.

The solution set is {x | x ≥ 7/3 or x ≤ 1} and its graph is