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Is x < 5?

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Is x < 5?

by Brent@GMATPrepNow » Thu Aug 03, 2017 1:03 pm

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jjjinapinch wrote:Is x < 5?
(1) x² > 5
(2) x² + x < 5

Official Guide question
Answer: B
Target question: Is x < 5?

Statement 1: x² > 5
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 3 is a possible value since 3² > 5. In this case, x < 5
Case b: x = 6 is a possible value since 6² > 5. In this case, x > 5
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of testing values when a statement doesn't feel sufficient, read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: x² + x < 5
IMPORTANT: x² is greater than or equal to 0 for ALL values of x.
So, let's let 0+ represent the value of x². This notation reminds us that x² equals some value that's greater than or equal to 0
Our inequality becomes: (0+) + x < 5
In order for the left side to be less than 5, x CANNOT be greater than or equal to 5
In other words, it MUST be the case that x < 5
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Sat Aug 26, 2017 11:47 am, edited 1 time in total.
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by [email protected] » Thu Aug 03, 2017 9:03 pm

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Hi jjjinapinch,

We're asked if X is less than 5. This is a YES/NO question. We can answer this question by TESTing VALUES and paying careful attention to the question that is asked. To start, we know that ANY negative value for X and X=0 will lead to a YES answer (as well as any values of X that are less than 5). Answers that are 5 or greater will lead to a NO answer.

1) X² > 5

With this inequality, there are clearly multiple values for X (positive and negative).
IF... X = 5, then the answer to the question is NO.
IF... X = -5, then the answer to the question is YES.
Fact 1 is INSUFFICIENT

2) X² + X < 5

With this inequality, we know that there are negative solutions and that X=0 is a solution, so there are clearly "NO" answers to this question. However, we now have to think about how big X could get. Thankfully, there's a way to avoid doing that exact math and use a bit of 'brute force' to help us to stop working...

IF.... X=1, then (X^2 + X) = 1^2 + 1 = 2... which is less than 5
IF.... X=2, then (X^2 + X) = 2^2 + 2 = 6... which is GREATER than 5
As X increases, the sum of (X^2 + X) will also increase. Since we're already greater than 5 when X=2, there's no reason to check any larger values. Given this inequality, X MUST be less than 2, so the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

Final Answer: B

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by Jay@ManhattanReview » Fri Aug 25, 2017 1:00 am

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jjjinapinch wrote:Is x < 5?
(1) x^2 > 5
(2) x^2 + x < 5

Official Guide question
Answer: B
Nice solutions by Brent and Rich. Here's one with brute force.

S1: x^2 > 5
  • Plug-in a value of x > 5, say x = 6 such that x^2 > 5. We see that (6^2 = 36) > 5. The answer is No.

    Plug-in a value of x < 5, say x = -6 such that x^2 > 5. We see that (-6)^2 > 5 => 36 > 5. The answer is Yes.
No unique answer!

S2: x^2 + x < 5
  • Plug-in a value of x > 5, say x = 6 such that x^2 + x > 5. We see that (6^2 + 6 = 42) < 5. It's not a valid value. Let's try with a smaller value.

    Say x = 2 such that x^2 + x > 5. We see that (2^2 + 2 = 6) < 5. It's not a valid value. Let's try with another smaller value.

    Say x = 1 such that x^2 + x > 5. We see that (1^2 + 1 = 6) > 5. It's a valid value. The answer is Yes.
There is no need to test further as we now know that x < 2, this means that x < 5. Sufficient.

The correct answer: B

Hope this helps!

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by Scott@TargetTestPrep » Fri Nov 17, 2017 12:09 pm

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jjjinapinch wrote:Is x < 5?
(1) x^2 > 5
(2) x^2 + x < 5
We need to determine whether x < 5.

Statement One Alone:

x^2 > 5

If x = 3, then x is less than 5. However, if x = 6, then x is not less than 5. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x^2 + x < 5

Thus, we have x < 5 - x^2. Since x^2 is nonnegative, we have 5 - x^2 ≤ 5. Since x < 5 - x^2 and 5 - x^2 ≤ 5, we have x < 5.

Answer: B

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