1. **State the problem:** Given that $a > 0$ and $b < 0$, determine which of the following expressions is always negative: (a) $|ab|$ (b) $a |b|$ (c) $|a| b$ (d) $a + |b|$ 2. **Analyze each option:** - (a) $|ab|$: Since $a > 0$ and $b < 0$, the product $ab$ is negative. However, the absolute value $|ab|$ is always non-negative (zero or positive). So, $|ab|$ is never negative. - (b) $a |b|$: Here, $a > 0$ and $|b|$ is the absolute value of $b$, which is positive since $b < 0$. Therefore, $a |b|$ is a product of two positive numbers, so it is positive, not negative. - (c) $|a| b$: Since $a > 0$, $|a| = a > 0$. Given $b < 0$, the product $|a| b = a b$ is positive times negative, which is negative. So, $|a| b$ is always negative. - (d) $a + |b|$: Since $a > 0$ and $|b| > 0$, their sum $a + |b|$ is positive, not negative. 3. **Conclusion:** The only expression that is always negative is (c) $|a| b$. **Final answer:** (c) $|a| b$ is always negative.