Which of the Following Expressions CANNOT have a Negative Value? GMAT Problem Solving

Question: Which of the following expressions CANNOT have a negative value?

1. |a + b| – |a – b|
2. |a + b| – |a|
3. |2a + b| – |a + b|
4.  a^2 + b^2 – 2|ab|
5. |a^3 + b^3| – a – b

“Which of the following expressions CANNOT have a negative value?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

We can use number approach to solve this

1. Choose a negative A for the first one to make the second absolute value expression smaller and the expression as a whole negative.
2.  In this case, A can also be turned negative.
3.  It is recommended to utilise a negative A (-1) and a B value of 2 for this problem.
4.  correct
5. Here, you can use 1/2 and 1/2 for A and B since fractions will get smaller as they go closer to zero when they are cubed. Then, from these smaller terms, the original fractions are deducted.

 a^2 + b^2 – 2|ab| is correct. 

Correct Answer: D

Approach Solution 2:

There is another approach to solve this question which is pretty easy

See that, \(a^2+b^2−2|ab| = (|a|−|b|)^2\)

Any number's square has a value higher than or equal to 0.
Therefore answer is D
Think about scenarios where negative expressions will occur, and you can eliminate.
In each instance, we must reduce the first term relative to the second term.

1.  |a + b| – |a – b|
   The absolute value of (a - b) will be greater than that of (a + b) if an is negative and b is positive. Therefore, this statement will be negative.
2. |a + b| – |a|
   If b is negative and a positive, (a + b) will have a lower absolute value than a. This phrase will therefore be negative.
3. |2a + b| – |a + b|
   (A + B) would be more than (2A + B) if an is negative and b is positive (greater than 2A). Therefore, this statement will be negative.

5. |a^3 + b^3| – a – b
   = |a^3 + b^3| – (a + b)
   (A + B) would be more than (2A + B) if an is negative and b is positive (greater than 2A). Therefore, this statement will be negative.

a^2 + b^2 – 2|ab| is correct 

Correct Answer: D

Approach Solution 3:

We can resolve this using a numerical technique.

To make the second absolute value expression smaller and the expression as a whole negative, choose a negative A for the first one.
A is also capable of becoming negative in this situation.
For this issue, it is advised to use a negative A (-1) and a B value of 2.
correct
Since fractions will grow smaller as they approach closer to zero when they are cubed, you may use 1/2 and 1/2 for A and B in this situation. After that, the initial fractions are subtracted from these smaller terms.

a^2 + b^2 – 2|ab| is correct 

Correct Answer: D

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