If |x + 1 | > 2x - 1, Which Of The Following Represents The Correct Range GMAT Problem Solving

Question: If |x + 1 | > 2x - 1, which of the following represents the correct range of values of x?

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

“ If |x + 1 | > 2x - 1, which of the following represents the correct range”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. GMAT quant questions are designed to examine the rational and logical skills of the candidates. The candidates must know the mathematical knowledge to interpret graphic data, arithmetic and algebra and to solve the GMAT Problem Solving questions. The GMAT Quant topic in the problem-solving part cites certain calculative mathematical questions that can be cracked only by suitable qualitative skills. The candidate has to do proper mathematical calculations in order to choose the correct answer for this GMAT quant question.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• |x + 1 | > 2x - 1

Find out:

• The correct range of values of x

From the given equation |x + 1 | > 2x - 1, we can get,

=> x+1 > 2x-1 [ since | x+1 |= x + 1, when (x+1) ≥ 0 or, x ≥ −1 ]
Or, 2x-1 < x+1
Or, x < 2
Therefore, x≥ −1 and x< 2 ….(i)

It is given that |x + 1 | > 2x - 1
Therefore, we get,
-(x+1) > 2x-1 [ |x+1|= -(x+1), when (x+1) < 0 Or, x< -1 ]
Or, 2x-1< -(x+1)
Or, 2x-1< -x-1
Or, 3x < 0
Or, x < 0
Therefore, x< −1 and x< 0 ….(ii)

Combining equations (i) and (ii), we get,
(x< −1 and x< 0) or (x≥ −1 and x< 2)
(x< −1 and x< 0) or (x≥ −1 and x< 2)
Therefore, x< 2

Hence, B is the correct answer.

Correct Answer: B

Approach Solution 2:

The problem statement declares that:

Given:

• |x + 1 | > 2x - 1

Find out:

• The accurate range of values of x

We have the equation, |x + 1 | > 2x - 1,
If we set a condition when x is positive,
Therefore, we get |x + 1 | ≥ 0
=> x≥-1

Now it is required to check if the equation satisfies the given condition:
|x + 1 | > 2x - 1
=> x+1>2x-1
=> x<2

Therefore, we get the value of x less than 2 when x is a positive integer and x ≥-1.
Hence the value of x could be 0.1, 0.2, etc inclusive 1.99)

If we set a condition when x is negative,
Therefore, we get, |x + 1 | < 0
=> x+1<0
=> x <-1

It is required to check if the equation satisfies the given condition:
-|x + 1 | > 2x - 1
=> -x-1>2x-1
=> x <0

Therefore, the value of x is less than 0 as per the condition x is a negative number and x <-1.
Hence the value of x could be -1.1, -1.2, -3, -100 etc )

By merging both conditions,
Therefore, we can say, x < 2 as per condition, X is positive and x ≥-1 ( hence x could be 0.1, 0.2, etc inclusive 1.99 )
Therefore, we can say, x < 0 as per condition, X is negative and x <-1 ( hence x could be -1.1, -1.2, -3, -100 etc )

On combining the value of x could be 0.1, 0.2, etc inclusive 1.99 and X could be -1.1, -1.2, -3, -100 etc

Let’s observe the answer options

1. x < 0

This option is INCORRECT since the value of x could be 0.1, 0.2, etc inclusive 1.99.

2. x < 2

This option is CORRECT since the value of x could be 0.1, 0.2, etc inclusive 1.99 and the value of x could be -1.1, -1.2, -3, -100 etc

100. -2 < x < 0

This option is INCORRECT since X could be -1.1, -1.2, -3, -100, etc

500. -1 < x < 2

This option is INCORRECT since the value of x could be -1.1, -1.2 , -3, -100, etc

5. 0 < x < 2

This option is INCORRECT as X could be -1.1, -1.2 , -3, -100, etc

Hence, B is the correct answer.

Correct Answer: B

Approach Solution 3:

The problem statement declares that:

Given:

• |x + 1 | > 2x - 1

Find out:

• The exact range of values of x

We can get the correct range of the values of x by determining the answer choices properly.

Let’s scan the answer options

It can be noticed that there are some choices of answers that say x=1 is a solution.

Some choices of answers say x=1 is not a solution.

Hence, let’s examine the value of x = 1

As per the given equation, |x + 1 | > 2x - 1 we can get,
|1 + 1 | > 2(1) - 1
By simplifying, we can get, 2 > 1
Therefore, x=1 is an answer to the inequality.

Since the answer options, A and C do not imply x=1 as an answer, thereby they get eliminated.

Now it is required to scan the other options (B, D and E)
It can be noticed that there are some choices of answers that say x= -1 is a solution.
Some choices of answers say x= -1 is not a solution.
Hence, let’s examine the value of x = -1

As per the given equation, |x + 1 | > 2x - 1 we can get,
|(-1) + 1 | > 2(-1) - 1
By simplifying, we can get, 0 > -3
Therefore, x= -1 is an answer to the inequality.

Since the answer options, D and E do not indicate x= -1 as an answer, thereby they get eliminated.

Hence, B is the correct answer.

Correct Answer: B

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