Key GMAT Strategy: Rephrasing the Target Question

Brent Hanneson - October 8, 2021

To set up this article try solving the following question:

Target question: Is x2 + 6x + 5 ≤ -4?

Statement 1: x < 10

Statement 2: x > -3

After reading the target question and statement 1, did you immediately start testing values of x that are less than 10?

If so, you may have started with a value like x = 9, in which case x2 + 6x + 5 = 92 + 6(9) + 5 = 140. So, the answer to the target question is NO.

Similarly, if x = 5, then x2 + 6x + 5 = 52 + 6(5) + 5 = 60, in which case the answer to the target question is NO.

If x = 0, then x2 + 6x + 5 = 02 + 6(0) + 5 = 5, in which case the answer to the target question is NO.

If x = -1, then x2 + 6x + 5 = (-1)2 + 6(-1) + 5 = 0, in which case the answer to the target question is NO.

And so on....

After testing a few values of x, one might feel confident that statement 1 is sufficient since we keep getting the same answer to the target question.

Unfortunately, it’s not sufficient. Here’s why:

First notice that we can rephrase the target question by adding 4 to both sides of the inequality to get: Is x2 + 6x + 9  0?

Aside: This step is a must-have strategy in your mathematical toolbox, since we can often learn a lot about a quadratic equation/inequality by setting it equal to zero.

From here, notice that we can factor the left side to get: Is (x + 3)2  ≤ 0?

At this point, we should ask “What value(s) of x will satisfy the inequality (x + 3)2  ≤ 0?”

Well, since the square of any number is always greater than or equal to zero, we can see that (x + 3)2 can never be negative. However, if x = -3, then (x + 3)2 = 0

So, x = -3 is the only x-value that satisfies the inequality (x + 3)2 ≤ 0, which means we can rephrase the target question one last time to get: Does x = -3?

Aside: Given the GMAT’s significant time constraints, many students don’t bother to rephrase the target question because they’re in a hurry to start analyzing the statements immediately. However, we can often analyze the statements much faster by spending a little time rephrasing (i.e., simplifying) the target question first.  

Okay, now that we’ve rephrased target question to get Does x = -3?, notice how quickly we can handle the two statements.

Statement 1: x < 10

If x < 10, then x could equal 5, in which case the answer to the rephrased target question is “NO, x does not equal -3”

However, x could also equal -3, in which case the answer to the rephrased target question is “YES, x equals -3” This means statement 1 is not sufficient.

Statement 2: x > -3

If x > -3, then x can never equal -3, which means the only possible answer to the rephrased target question is “NO, x does not equal -3.”

Therefore, statement 2 is sufficient.

Answer: B

 

Be sure to practice this must-have strategy before test day.

To get you started, here’s a video on rephrasing the target question, and here are several examples in which rephrasing the target question will make your analysis much easier/faster:

1. Target question: If x > y, is cx > cy?

Rephrased target question:  Is c > 0? Explanation

 

2. Target question: If 2.00x and 3.00y are 2 numbers in decimal form with thousandths digits x and y, is 3(2.00x) > 2(3.00x)?

Rephrased target question:  Is 3X > 2Y? Explanation

 

3. Target question: If n is an integer, is (0.1)n greater than (10)n?

Rephrased target question:  Is n negative? Explanation

 

4. Target question: Does a² + b²= 2ab?

Rephrased target question: Does a = b? Explanation

 

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