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If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the

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Bunuel
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If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] New post   23 Apr 2024, 01:30
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­If \(x\) is a non-zero integer and \(|6- 3x| +3|x + 2| = 12\), what is the value of \(x^4 - 5x^2 + 4\)?

A. -12
B. 0
C. 4
D. 6
E. 12­


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gmatophobia
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If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] New post   23 Apr 2024, 02:35
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Bunuel wrote:
­If \(x\) is a non-zero integer and \(|6- 3x| +3|x + 2| = 12\), what is the value of \(x^4 - 5x^2 + 4\)?

A. -12
B. 0
C. 4
D. 6
E. 12­
 

The critical points are -2 and 2.

Let's plot these over the number line

---------- Region 1 --------- -2 ---------- Region 2 --------- 2 ---------- Region 3 ---------

In Region 1 (\(x < -2\))

\(|6- 3x| +3|x + 2| = 12\)
­
\((6- 3x) -3(x + 2) = 12\)

\(6 - 3x -3x - 6 = 12\)

\(x = -2\)  → No valid point

In Region 2 \((-2 \leq x \leq 2)\)

\(|6- 3x| +3|x + 2| = 12\)
­
\((6- 3x) +3(x + 2) = 12\)

\(6 - 3x -3x + 6 = 12\)

\(12 = 12\)­ → x = {-2, -1, 1, 2}

In Region 3 (\(x > 2\))

\(|6- 3x| +3|x + 2| = 12\)
­
\(-6 + 3x + 3x + 6= 12\)

\(x = 2 \)

\(x = 2\)­ → No valid point­

Hence, \(x\) can be {-2, -1 , 1, 2} → For all values of x, \(x^4 - 5x^2 + 4\) = 0

Option B­
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Re: If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] New post   23 Apr 2024, 02:45
[quote][/quote]

gmatophobia

Is there any post where you have explained this critical points method?
Coz I took 4 scenarios of (+,+ ; +,-; -- ; -+) - to test equation, before getting the values
Which was a lil time-consuming
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If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] New post   23 Apr 2024, 03:29
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Shwarma wrote:
Quote:
 

gmatophobia

Is there any post where you have explained this critical points method?
Coz I took 4 scenarios of (+,+ ; +,-; -- ; -+) - to test equation, before getting the values
Which was a lil time-consuming

­Hi Shwarma

You may want to refer the below posts 

https://gmatclub.com/forum/inequations- ... 54664.html

https://gmatclub.com/forum/inequations- ... 54738.html
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Re: If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] New post   23 Apr 2024, 06:02
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­\(|6−3x|+3|x+2|=12\)

\(|3(2−x)|+3|x+2|=12 \)

Looking at the equation one notices that the equation is adding two multiples of 3 to get 12. The two pairs of multiples of 3 which yield 12 are: \((12+0)\), \((9+3)\) and \((6+6)\).

Logically one can deduce that \(x\) can never be \(3\) or greater as \(3|x+2| \) on its own will always exceed 12. Similarly, \(x\) can never be smaller than \(-3\) as then \(|3(2−x)|\) alone will exceed 12. This leaves one with 5 numbers: \(-2, -1, 0, 1, 2\), however, the stem tells us that \(x\) is a non-zero integer. Plugging in any of \(-2, -1, 1, 2\) into \(|3(2−x)|+3|x+2|=12 \) will hold. Similarly, plugging in of these values for \(x\) into \(x^4−5x^2+4\) will always yield \(0\). 

ANSWER B

 
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Re: If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink]
23 Apr 2024, 06:02
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