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# If x is an integer and (x2 -16)(x2 + x -2) < 0, what is the sum of all

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Manager
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If x is an integer and (x2 -16)(x2 + x -2) < 0, what is the sum of all  [#permalink]

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27 Nov 2018, 12:40
4
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Difficulty:

85% (hard)

Question Stats:

42% (01:42) correct 58% (01:45) wrong based on 51 sessions

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If x is an integer and ($$x^2$$ - 16)($$x^2$$ + x - 2) < 0, what is the sum of all the possible values of x?

A. -1
B. 0
C. 1
D. 2
E. 3
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Joined: 12 Mar 2018
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Location: Iran (Islamic Republic of)
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WE: Management Consulting (Energy and Utilities)
Re: If x is an integer and (x2 -16)(x2 + x -2) < 0, what is the sum of all  [#permalink]

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27 Nov 2018, 22:50
rencsee wrote:
If x is an integer and ($$x^2$$ - 16)($$x^2$$ + x - 2) < 0, what is the sum of all the possible values of x?

A. -1
B. 0
C. 1
D. 2
E. 3

To solve such an inequality, one should factorize left hand side; we can rewrite the left hand side as:
$$(x-4)(x+4)(x+2)(x-1)<0$$
so, the roots are: -4, -2, +1 & +4.
As you can see, x=-100 yields a positive (+ve) value because of 4 negative values resulted from each parenthesis, while we are looking for a -ve value.

According to attached diagram we can see that integer -ve values are -3, +2 & +3. So the sum of these values equal +2.
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Math Expert
Joined: 02 Aug 2009
Posts: 7765
Re: If x is an integer and (x2 -16)(x2 + x -2) < 0, what is the sum of all  [#permalink]

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27 Nov 2018, 22:50
rencsee wrote:
If x is an integer and ($$x^2$$ - 16)($$x^2$$ + x - 2) < 0, what is the sum of all the possible values of x?

A. -1
B. 0
C. 1
D. 2
E. 3

Let us factorize ($$x^2$$ - 16)($$x^2$$ + x - 2) < 0..
(x+4)(x-4)(x+2)(x-1)<0....
let us write the terms in ascending order..
$$(x+4)(x+2)(x-1)(x-4)<0$$..
Now, there are 4 terms and for their product to be NEGATIVE, Odd number should be negative.. so 1 or 3.
Critical points now
(I) x<-4...
all 4 terms negative....NO
(II) -4<x<-2..
x+4 is positive and rest 3 terms negative.. YES
possible value in range -2<x<-4 is -3
(III) -2<x<1
two terms positive and two terms negative.. NO
(IV) 1<x<4
Only x-4 is negative.... YES
possible values in range 1<x<4 is 2 and 3.
(V) x>4
all positive....NO

so all in all possible values are -3, 2, 3 and their sum = 2+3-3=2
D
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Posts: 223
Re: If x is an integer and (x2 -16)(x2 + x -2) < 0, what is the sum of all  [#permalink]

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27 Nov 2018, 23:02
(x+4)(x+2)(x-1)(x-4) < 0
-4<x<-2 || 1<x<4

-3 + 2 + 3
= 2
Ans D
Re: If x is an integer and (x2 -16)(x2 + x -2) < 0, what is the sum of all   [#permalink] 27 Nov 2018, 23:02
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