Q.144. Solution of inequality, Domain.

Question 144.
Solve for x: (2x + 14) / (5x - 1) ≤  x + 3.

Answer 144.
(2x + 14) /(5x - 1) ≤ x + 3

Case 1:
If 5x - 1 > 0,i.e., x > 1/5, then
2x + 14 ≤ (5x - 1)(x + 3)
=> 2x +14 ≤ 5x^2 + 14x - 3
=> 5x^2 +12x - 17 ≥ 0
=> (5x + 17)(x - 1) ≥ 0
=> 5x +17 ≥ 0 and x - 1 ≥ 0 OR 5x +17 ≤ 0 and x - 1 ≤ 0 under the condition that x > 1/5
=> x ≥ - 17/5 and x ≥ 1 OR x ≤ - 17/5 and x ≤ 1 under the condition that x > 1/5
=> x ≥ 1.

Case 2:
If 5x - 1 < 0,i.e., x < 1/5, then
2x + 14 ≥ (5x - 1)(x + 3)
=> 2x +14 ≥ 5x^2 + 14x - 3
=> 5x^2 +12x - 17 ≤ 0
=> (5x + 17)(x - 1) ≤ 0
=> 5x +17 ≥ 0 and x - 1 ≤ 0 OR 5x +17 ≤ 0 and x - 1 ≥ 0 under the condition that x < 1/5
=> x ≥ - 17/5 and x ≤ 1 OR x ≤ - 17/5 and x ≥ 1 under the condition that x < 1/5
=> - 17/5 ≤ x < 1/5.

=> x belongs to [- 17/5, 1/5) U [1, ∞).

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