Question

Solve the following system of inequalities:

`x/(2x + 1) ≥ 1/4, (6x)/(4x - 1) < 1/2`

Answer 1

From the first inequality, We have `x/(2x + 1) - 1/4 ≥ 0`

⇒ `(2x - 1)/(2x + 1) ≥ 0`

⇒ (2x – 1 ≥ 0 and 2x + 1 > 0) or (2x – 1 ≤ 0 and 2x + 1 < 0)   ......[Since 2x + 1 ≠ 0)

⇒ `(x ≥ 1/2 "and"  x > - 1/2)` or `(x < 1/2 "and"  x < - 1/2)`

⇒ `x ≥ 1/2` or `x < -1/2`

⇒ x ∈ `(-oo, - 1/2) ∪ [1/2, oo)`  ....(1)

From the second inequality, We have `(6x)/(4x - 1) - 1/2 < 0`

⇒ `(8x + 1)/(4x - 1) < 0`

⇒ (8x + 1 < 0 and  4x – 1 > 0) or (8x + 1 > 0 and 4x – 1 < 0)

⇒ `(x < - 1/8 "and" x > 1/4)` or `(x > 1/8 "and"  x < 1/4)`

⇒ x ∈ `(-1/8, 1/4)`  (Since the first is not possible)  ...(2)

Note that the common solution of (1) and (2) is null set. Hence, the given system of inequalities has no solution.