Question

If \( x>y \) and \( z<0 \), then:

• A. \( xz>yz \)
• B. \( xz \geq yz \)
• C. \( \frac{x}{z}>\frac{y}{z} \)
• D. \( \frac{x}{z}<\frac{y}{z} \)

Hint:

When dividing inequalities by a negative number, always reverse the direction of the inequality. This rule is crucial for handling problems involving negative values.

Answer 1

The Correct Option is D

Step 1: For \( z<0 \), dividing both \( x>y \) by \( z \) reverses the inequality. Therefore, we have: \[ \frac{x}{z}<\frac{y}{z}. \] Step 2: To verify, consider any example where \( x>y \) (e.g., \( x = 5, y = 3 \)) and \( z<0 \) (e.g., \( z = -2 \)). Then: \[ \frac{x}{z} = \frac{5}{-2} = -2.5 \quad {and} \quad \frac{y}{z} = \frac{3}{-2} = -1.5. \] Clearly, \( \frac{x}{z}<\frac{y}{z} \).