Linear Inequations-8

Solve the system of inequaitons: x/(2x + 1) ≥ 1/4 , 6x/(4x - 1) < 1/2

Solution: The given system of inequations is

x/(2x + 1) ≥ 1/4 ---------------------------------------- (i)

6x/(4x – 1) < 1/2 --------------------------------------------- (ii)

Now, x/(2x + 1) ≥ 1/4 => x/(2x + 1) – 1/4 ≥ 0

=>[4x – (2x + 1)]/4(2x + 1) ≥ 0

=> (2x - 1)/(2x + 1) ≥ 0 [Multiplying both sides by 4]

=>x ∈ (-∞, -1/2) U [1/2, ∞)

Thus the solution set of the inequation (i) is ∈ (-∞, -1/2) U [1/2, ∞)---------(iii)

<(-∞) ---------------------------(-1/2)-------------(1/2)---------------------------(+∞) >

And, 6x/(4x - 1) < 1/2

=>6x/(4x - 1) – 1/2 < 0

=> [12x – (4x - 1)]/2(4x - 1) < 0

=> (8x + 1)/2(4x - 1) < 0

(8x + 1)/(4x - 1) < 0 [Multiplying both sides by 2]

=>x ∈ {-1/8, 1/4}

Thus, the solution set of inequation (ii) is (-1/8, 1/4)

x ∈ {-1/8, 1/4}------------------------------------------------------------------------(iv)

< (-∞)-----------------------(-1/8)----------------(1/4)------------------------------(+∞)>

It is evident from the figure that the intersection of (iii) and (iv) is the null set.

Hence, the given system of inequations has no solution.