Question: Solve for x: 0 < |x| – 4x < 5 = ?

1. x < 0
2. 0 < x < 1
3. -3/5 < x < 1
4. -3/5 < x < 0
5. -1 < x < 0

Correct Answer: E
Solution and Explanation:
Approach Solution 1:

The problem statement asks to find out the expression in terms of x where the equation is 0 < |x| – 4x < 5.
Therefore we can say,
1st condition: 5 + 4x > |x|
Or, 5 + 4x > x
Or, 5 > - 3x
Or, x > – 5/3

On the other hand, we can say: 5 + 4x > -x
Or, x > -1

2nd condition: 4x < |x|
Or, 4x < x
Or, x < 0

On the other hand, we can say: 4x < -x
Or, x < 0

Therefore, the range lies between -1 to 0 i.e -1 < x < 0

Approach Solution 2:

The problem statement asks to find out the expression in terms of x where the equation is 0 < |x| – 4x < 5.
Let’s consider option A i.e x < 0
If the value of x = -1, then |x| – 4x = 5 does not satisfy the inequality. Hence, option A gets eliminated.
By analysing option B i.e. 0 < x < 1 and option C i.e. -3/5 < x < 1
If we assume, x = ½, then |x| - 4x is negative. Therefore, options B and C get eliminated.
If the value of x = -⅘, which satisfies the equation 0 < |x| – 4x < 5 since |x| – 4x = 4.
However, -⅘ does not lie in the range between -3/5 < x < 0. Therefore option D is out.

Hence, option E is the answer.

Approach Solution 3:

The problem statement asks to find out the expression in terms of x where the equation is 0 < |x| – 4x < 5.

If x > 0, then 0 < x – 4x < 5
=> 0 < – 3x < 5
=>0 < x < – 5/3
=>-5/3 < x < 0

Therefore, it is not valid since the value of x does not lie within the range of x > 0.

OR
If x < 0, then 0 < (-x) – 4x < 5
=> 0 < – 5x < 5
=>0 < x < – 1
=> -1 < x < 0

Therefore, it is valid as the value of x lies within the range of x < 0

There exists only one proper solution i.e -1< x < 0.

“Solve for x: 0 < |x| – 4x < 5 = ?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide 2018”. The GMAT Problem Solving questions test the candidates’ skills in estimating numerical problems accurately. GMAT Quant practice papers help the candidates to get familiarised with varieties of questions that will improve their learning.

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