GMAT CLUB OLYMPICS: If x is a non-zero integer, what is the value

Official Solution:


If \(x\) is a non-zero integer, what is the value of \(x^4 - 5x^2 + 4\)?

This one is tricky!!!

(1) \(|6- 3x| +3|x + 2| = 12\)

Reduce by 3: \(|x - 2| + |x + 2| = 4\)

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

Next, when \(-2 \leq x \leq 2\), then \(|x - 2|=-(x-2)\) and \(|x + 2|=x+2\). So, \(|x - 2| + |x + 2| = 4\) will become \(-(x - 2) + (x + 2) = 4\), which gives \(4=4\). This means 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 \(x\) is -2, -1, 1, or 2. For each of these values of \(x\), the value of \(x^4 - 5x^2 + 4\) is 0. Sufficient.

(2) \(|x^2 - 2| - |x - 2| = -2\)

Re-arrange: \(|x^2 - 2| +2= |x - 2| \).

Notice here that if \(x\) is a positive integer or 0, then \(|x^2 - 2| \) is more than or equal to \(|x - 2|\) (they are equal when \(x=1\) and \(x=0\) ). So, \(|x^2 - 2| +2\) will be greater than \(|x - 2| \). Thus, \(x\) cannot be positive.

Now, when \(x < 0\), then \(|x - 2|=-(x-2)=2-x\). In this case we'd have \(|x^2 - 2| - (2-x)= -2\). This gives: \(|x^2 - 2| =x\). Square both side: \(x^4 - 4x^2+4 =x^2\).

Re-arrange: \(x^4 - 5x^2+4 =0\). That is exactly what we were asked to find. Sufficient.


Answer: D