# Inequality – Problems

## Inequality – Problems

Example 1: If the |x + 2| ≤ 9 then

Solution: Given that |x + 2|≤ 9

-9 ≤ (x + 2) ≤ 9

-9 – 2 ≤ x + 2 – 2 ≤ 9 – 2

-11 ≤ x ≤ 7

x є [7, -11]

Example 2: The solution set of (2x – 1)/ 3 ≥ (3x – 2)/ 4 – (2 – x)/ 5 is

Solution: Given that (2x – 1)/ 3 ≥ (3x – 2)/ 4 – (2 – x)/ 5

(2x – 1)/ 3 ≥ (15x – 10 – 8 + 4x)/ 20

(2x – 1)/ 3 ≥ (19x – 18)/ 20

20(2x – 1)/ 3 ≥ (19x – 18)

40x – 20 ≥ 3 (19x – 18)

40x – 20 ≥ 57x – 54

40x – 57x ≥ – 54 + 20

-17x ≥ -34

17x ≤ 34

x ≤ 34/ 17

x ≤ 2

x є (- ∞, 2]

Example 3: The solution set of (|x – 2| – 1)/ (|x – 2| – 2) ≤ 0 is

Solution: Given that (|x – 2|- 1)/ (|x – 2| – 2) ≤ 0

Let |x – 2| = k

Given equation becomes

(k – 1)/ (k – 2) ≤ 0

(k – 1) (k – 2)/ (k – 2)² ≤ 0

(k – 1) (k – 2) ≤ 0

1 ≤ k ≤ 2 … (1)

1 ≤ |x – 2|≤ 2

Case (i): When 1 ≤ |x – 2|

|x – 2| ≥ 1

x – 2 ≥ 1 and x – 2 ≤ -1

x ≥ 3 and x ≤ 1

Case (ii): When |x – 2| ≤ 2

x – 2 ≤ 2 and x – 2 ≥ -2

x ≤ 4 and x ≥ 0 … (2)

From equation (1) and equation (2)

x є [0, 1] ∪ [3, 4].