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].