Question -
Answer -
Given:
(|x + 2| – x) / x < 2
Let us rewrite the equation as
|x + 2|/x – x/x < 2
|x + 2|/x – 1 < 2
By adding 1 on both sides, we get
|x + 2|/x – 1 + 1 < 2 + 1
|x + 2|/x < 3
By subtracting 3 on both sides, we get
|x + 2|/x – 3 < 3 – 3
|x + 2|/x – 3 < 0
Clearly it states, x ≠ -2 so two case arise:
Case1: x + 2 > 0
x > –2
In this case |x+2| = x + 2
x + 2/x – 3 < 0
(x + 2 – 3x)/x < 0
– (2x – 2)/x < 0
(2x – 2)/x < 0
Let us consider only the numerators, we get
2x – 2 > 0
x>1
x ϵ (1, ∞) ….(1)
Case 2: x + 2 < 0
x < –2
In this case, |x+2| = – (x + 2)
-(x+2)/x – 3 < 0
(-x – 2 – 3x)/x < 0
– (4x + 2)/x < 0
(4x + 2)/x < 0
Let us consider only the numerators, we get
4x + 2 > 0
x > – ½
But x < -2
From the denominator we have,
x ∈ (–∞ , 0) …(2)
From (1) and (2)
∴ x ∈ (–∞ , 0) ⋃ (1, ∞)