For all nonzero values of a and b, the value of which of the following expressions is always negative?
The correct answer is E.
The |x| means the absolute value of x and, by definition,
|x| = x if x ≥ 0 and |x| = -x if x < 0.
In any case, |x| is positive for all nonzero values of x.
Because the sum of 2 positive numbers is positive, |a| + |b| > 0.
Multiplying both sides of this inequality by -1 and remembering to switch the direction of the inequality sign, we get
-(|a| + |b|) = -|a| - |b| < -0 = 0.
Therefore, the expression in E is always negative.