Solve: |-9 + |d + 7| + |-3d - 2|| < 0 - Complex Absolute Value Inequality

Because absolute values are always ≥ 0! The smallest possible values are $|d + 7| = 0$ and $|-3d - 2| = 0$, making the minimum of the entire expression -9 + 0 + 0 = -9. Since -9 is not less than 0, the inequality is impossible.

Set each absolute value to zero (their smallest possible value). For $-9 + |d + 7| + |-3d - 2|$, the minimum is when both absolute values equal 0, giving us -9 + 0 + 0 = -9.

Then we'd have solutions! Since the minimum value is -9, the expression can be greater than 0. We'd need to find when $-9 + |d + 7| + |-3d - 2| > 0$, or when $|d + 7| + |-3d - 2| > 9$.

Yes! For complex absolute value inequalities, first determine the range of possible values for the left side. If this range doesn't overlap with what the inequality requires, you've saved yourself lots of work!

Never! Individual absolute values like $|x|$ are always ≥ 0. However, expressions containing absolute values (like $-9 + |x|$) can be negative if the constant term is large enough.