HOT Competition 1 Sep/8AM: If x 5 and x is a non-zero integer, what

Official Solution:


If \(x ≠ 5\) and \(x\) is a non-zero integer, what is the value of \(\frac{|x| + 5}{5 - x}\) ?

If \(x > 0\), then \(|x|=x\) and \(\frac{|x| + 5}{5 - x}=\frac{x + 5}{5 - x}\)

If \(x < 0\), then \(|x|=-x\) and \(\frac{-x + 5}{5 - x}=1\)

So, if \(x\) is positive, then the value of \(\frac{|x| + 5}{5 - x}\) depends on exact value of \(x\) but if \(x\) is negative, then he value of \(\frac{|x| + 5}{5 - x}\) is 1.

(1) \(|y + 5| < 1 - x\):

The left hand side (\(|y+5|\)), is the absolute value of a number, so it must be positive or 0. Thus, \(0 < 1 - x\). This gives \(x < 1\). Since given that \(x\) is a non-zero integer, then \(x\) must be negative integer (-1, -2, ...). As we deduced above, if \(x\) is negative, then he value of \(\frac{|x| + 5}{5 - x}\) is 1. Sufficient.

(2) \(\frac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1\):

\(\frac{|x|}{x}=-1\);

\(|x|=-x\). This means that \(x\) is negative. As we deduced above, if \(x\) is negative, then he value of \(\frac{|x| + 5}{5 - x}\) is 1. Sufficient.


Answer: D