Solve the following system of inequalities: x2x+1≥14,6x4x-1<12 - Mathematics

Sum

Solve the following system of inequalities:

x/(2x + 1) ≥ 1/4, (6x)/(4x - 1) < 1/2

Solution

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.

Concept: Inequalities - Introduction
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APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 6 Linear Inequalities
Solved Examples | Q 8 | Page 103
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