Solution:

It is given to us that : (|x + 2| – x) / x < 2

We can rewrite the above equation as:

|x + 2|/x – x/x < 2

|x + 2|/x – 1 < 2

Upon adding 1 to both sides, we get

|x + 2|/x – 1 + 1 < 2 + 1

|x + 2|/x < 3

Upon subtracting 3 from both sides, we get

|x + 2|/x – 3 < 3 – 3

|x + 2|/x – 3 < 0

The above equation clearly states that x ≠ -2 so two cases arise, which are:

Case1: x + 2 > 0

It implies that : x > –2

In this case, we have

|x+2| = x + 2

x + 2/x – 3 < 0

( x + 2 – 3x ) / x < 0

– (2x – 2)/x < 0

( 2x – 2)/x < 0

Considering only the numerators we get:

2x – 2 > 0

This implies that: x>1

Therefore, x ϵ (1, ∞) ….(1)

Case 2: x + 2 < 0

This implies that x < –2

In this case, we have

|x+2| = – (x + 2)

-(x+2)/x – 3 < 0

(-x – 2 – 3x)/x < 0

– (4x + 2)/x < 0

(4x + 2)/x < 0

Considering only the numerators, we get the following result:

4x + 2 > 0

x > – ½

But we know, x < -2

We have, from the denominator the following result:

x ∈ (–∞ , 0) …(2)

Using equations (1) and (2)

Therefore,  x ∈ (–∞ , 0) ⋃ (1, ∞)