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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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If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   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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Bunuel
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Re: If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   04 May 2024, 20:45
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Official Solution:

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


Reduce \(|6 - 3x| + 3|x + 2| = 12\) by 3 to get \(|x - 2| + |x + 2| = 4\).

Notice that if \(x\) is less than -2 or greater than 2, then \(|x - 2| + |x + 2|\) will be more than 4. For example, if \(x < -2\), then \(|x - 2| > 4\), and if \(x > 2\), then \(|x + 2| > 4\). Thus, \(x\) must be between -2 and 2, inclusive.

Next, when \(-2 \leq x \leq 2\), \(|x - 2| = -(x-2)\) and \(|x + 2| = x+2\). Therefore, \(|x - 2| + |x + 2| = 4\) simplifies to \(-(x - 2) + (x + 2) = 4\), which simplifies further to \(4 = 4\). This implies that ANY value of \(x\) such that \(-2 \leq x \leq 2\) satisfies \(|x - 2| + |x + 2| = 4\).

We are told that \(x\) is a non-zero integer; thus, \(x\) can only be -2, -1, 1, or 2. For each of these values, \(x^4 - 5x^2 + 4\) equals 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] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   23 Apr 2024, 02:35
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Bunuel
­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] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   23 Apr 2024, 02:45
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[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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Re: If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   23 Apr 2024, 03:29
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Shwarma
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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] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   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] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   04 May 2024, 21:01
Naw I solved the undercover quadratic equation first by setting it equally to zero. I got 1 and 4.

Substituted 1 and 4 back into the original equation and only 1 worked.

Posted from my mobile device
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If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink] If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the   28 May 2024, 07:08
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If \(x\) is a non-zero integer and \(|6- 3x| +3|x + 2| = 12\), what is the value of \(x^4 - 5x^2 + 4\)?

We see that there are only two constraints on the value of \(x\):

  \(x\) is a non-zero integer

  \(|6- 3x| +3|x + 2| = 12\)

So, ANY value of \(x\) that satisfies those constraints should get us to the correct answer. Thus, we can answer this question by finding just one value of \(x\) that works.

Simplify the equation.

\(|2 - x| +|x + 2| = 4\)

Try assuming that both expressions in the absolute value bars are nonnegative, meaning \(2 - x ≥ 0\) and  \(x + 2 ≥ 0\),  and solving.

\(2 - x - x + 2 = 4\)

\(4 = 4\)

So, any nonzero \(x\) such that \(2 - x ≥ 0\) and  \(x + 2 ≥ 0\) must work.

In other words, \(x\) is any nonzero integer such that \(−2 ≤ x ≤ 2\).

Try \(1\) in the expression.

\(1^4 - 5(1^2) + 4\)

\(1 - 5 + 4 = 0\)

A. -12
B. 0
C. 4
D. 6
E. 12­
 
Correct answer: B­
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If x is a non-zero integer and |6 - 3x| +3|x + 2| = 12, what is the [#permalink]
28 May 2024, 07:08
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