What is the value of integer x ?

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BTGmoderatorAT: Moderator; Posts: 426; Joined: Tue Aug 22, 2017 8:48 pmFollowed by: 1 member

by BTGmoderatorAT » Sun Sep 10, 2017 6:43 am

What is the value of integer x ?

(1) 4 < (x-1)*(x-1) < 16
(2) 4 < (x+1)*(x-1) < 16

OA C

Can some experts determine whether the statements are sufficient?

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Source: Beat The GMAT — Data Sufficiency | Sun Sep 10, 2017 6:43 am

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by [email protected] » Sun Sep 10, 2017 9:53 am

Hi ardz24,

We're told that X is an INTEGER. We're asked for the value of X. A bit of 'brute force' math can help you to work through this question relatively quickly.

1) 4 < (x-1)(x-1) < 16

Since X has to be an INTEGER, you can start at X=0 and 'work up' to find a solution. X cannot be 0, 1, 2 or 3 (since none of those results would fit the given range. However, X = 4 does fit - but no other positive integer value will fit that range. The prompt did not state that X had to be positive though, so we also have to consider NEGATIVE solutions. 'Working down' from 0, you'll find that X = -2 is also a solution (but no other negative integer solution exists).
Fact 1 is INSUFFICIENT

2) 4 < (x+1)(x-1) < 16

The same approach that we used in Fact 1 can be used here. You'll find that X = 3 and X = 4 are solutions and X = -3 and X = -4 are also solutions.
Fact 2 is INSUFFICIENT

Combined, there is only one answer that 'fits' BOTH Facts: X = 4
Combined, SUFFICIENT

Final Answer: C

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Rich

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What is the value of integer x ?

by GMATGuruNY » Sun Sep 10, 2017 11:18 am

ardz24 wrote:What is the value of integer x ?

(1) 4 < (x-1)*(x-1) < 16
(2) 4 < (x+1)*(x-1) < 16

Statement 1:
4 < (x-1)Â² < 16.
Implication:
x-1=Â±3, with the result that (x-1)Â²=9.
If x-1 = 3, than x=4.
If x-1 = -3, than x=-2.
Since x can be different values, INSUFFICIENT.

Statement 2:
4 < xÂ² - 1 < 16
5 < xÂ² < 17.
Implication:
xÂ²=9 or xÂ²=16.
Options for x:
Â±3, Â±4.
Since x can be different values, INSUFFICIENT.

Statements combined:
Of the options for x in Statement 1, only x=4 is among the options for x in Statement 2.
Thus, x=4.
SUFFICIENT.

The correct answer is C.

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Re: What is the value of integer x ?

by Scott@TargetTestPrep » Thu Jan 14, 2021 4:28 am

BTGmoderatorAT wrote: ↑
Sun Sep 10, 2017 6:43 am
What is the value of integer x ?

(1) 4 < (x-1)*(x-1) < 16
(2) 4 < (x+1)*(x-1) < 16

OA C

Can some experts determine whether the statements are sufficient?

Solution:

Statement One Alone:

4 < (x-1)*(x-1) < 16

Rewriting the inequality, we have:

4 < (x - 1)^2 < 16

Square rooting all 3 sides, we have:

2 < |x - 1| < 4

Since x is an integer, we see that |x - 1| = 3. That is, either x - 1 = 3 or x - 1 = -3. The former equation yields x = 4 while the latter yields x = -2. Statement one alone is not sufficient.

Statement Two Alone:

4 < (x+1)*(x-1) < 16

Rewriting the the inequality, we have:

4 < x^2 - 1 < 16

5 < x^2 < 17

Square rooting all 3 sides, we have:

√5 < |x| < √17

Since x is an integer, we see that |x| = 3 or |x| = 4. The former equation yields x = 3 or -3 while the latter yields x = 4 or -4. Statement two alone is not sufficient.

Statements One and Two Together:

Since the values of x in statement one could be 4 or -2 and the values of x in statement two could be 3, -3, 4 or -4, we see that x could only be 4 when we consider both statements together. The two statements together are sufficient.

Answer: C

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