solve x/2x+1 >=1/4 and 6x/4x-1 <1/2
Answers
Answer:
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.