What is the value of integer x ?

Question

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

Bunuel · Accepted Answer

Note: x is an integer. (1) 4<(x-1)*(x-1)<16 --> 4<(x-1)^2<16 --> (x-1)^2 is a perfect square between 4 and 16 --> there is only one perfect square: 9 --> (x-1)^2=9 --> x-1=3 or x-1=-3 --> x=4 or x=-2 . Two answers, not sufficient. (2) 4<(x+1)*(x-1)<16 --> 4 5 x^2 is a perfect square between 5 and 17 --> there are two perfect squares : 9 and 16 --> x^2=9 or x^2=16 --> x=3 or x=-3 or x=4 or x=-4 . Four answers, not sufficient. (1)+(2) Intersection of values from (1) and (2) is x=4 . Sufficient. Answer: C.

Hussain15 · Answer

Great explanation Bunuel!

Kudos!!

msand · Answer

Good question with a great explanation... Bunnel you are the champ :-)

zz0vlb · Answer

Bunuel, Just excellent!!!.

ksp · Answer

I have a doubt with that of choosing option 1.

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

4 < (x-1)^2 < 16

Taking square root.

2 < (x-1) < 4

This gives 2 scenarios. x>3 and x< 5

Since x is an integer, the option must be 4.

So the answer must be A rite .

Any inputs ??

KarishmaB · Answer

When you take the square root (which you can because all terms are positive), you get 
2 < |x - 1| < 4
If x is to be an integer, |x - 1| = 3
x can be 4 or -2

Remember,  \sqrt{x^2}
eq x  
Instead,  \sqrt{x^2}  = |x|

Bunuel · Answer

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HKHR · Answer

I tried solving the first inequality and got this:
Statement 1:
4 < (x-1)*(x-1) < 16

--> \(x^2 -2x -3 < 12\)

--> \(x^2 - 2x - 15<0\)

--> \((x+3) (x-5)<0\)

If x is an integer it means the function will be negative for the values \(x= -2, -1, 0, 1, 2, 3, 4.\)

What is the mistake here?

Bunuel · Answer

Why would you expand and manipulate 4 < (x-1)^2 < 16, when its already in its simplest form?

Anyway:
 4 < (x-1)^2 < 16 ;
 0 < x^2-2x-3 < 12  or  -12 < x^2-2x-15 < 0 ;
 -3<x<-1  or  3<x<5  -->  x=-2  or  x=4 .

Hope this helps.

sahilvijay · Answer

1) 4<(x-1)^2<16
3<x<5 or -3<x<-1
X=-2 or 4
A D eliminated

2) 4<x^2-1<16
5<x^2<16
x= -3,3,-4,4
B eliminated

Combine 1&2

X=4 is the answer

C is ANSWER

Posted from my mobile device

Kinshook · Answer

Asked: What is the value of integer x ?

(1) 4 < (x-1)*(x-1) < 16
4< (x-1)^2 <16; 
(x - 1)^2 = 9
x -1 = {-3,3}
x = {-2,4}
NOT SUFFICIENT

(2) 4 < (x+1)*(x-1) < 16
4< x^2 - 1<16
5<x^2<17
x^2 = {9,16}
x = {-4,-3,3,4}
NOT SUFFICIENT

(1) + (2)

(1) 4 < (x-1)*(x-1) < 16
4< (x-1)^2 <16; 
(x - 1)^2 = 9
x -1 = {-3,3}
x = {-2,4}

(2) 4 < (x+1)*(x-1) < 16
4< x^2 - 1<16
5<x^2<17
x^2 = {9,16}
x = {-4,-3,3,4}

Combining the results of (1) & (2) 
x = 4
SUFFICIENT

IMO C

Ahmed9955 · Answer

Hi  karishma ,
2 < |x - 1| < 4
can you tell , why have you not considered +/-2<|x-1|<+/-4 ?

Thanks

ScottTargetTestPrep · Answer

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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What is the value of integer x ?

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