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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
A. -12
B. 0
C. 4
D. 6
E. 12
22 Apr 2024, 07:31
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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
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
02 Mar 2025, 21:52
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I like the solution - it’s helpful.
