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: Given \( z < 0 \) and \( x > y \), dividing both sides of the inequality \( x > y \) by \( z \) (recall that \( z \) is negative) reverses the inequality sign. Hence,

\[ \frac{x}{z} < \frac{y}{z}. \]

Step 2: To verify this, consider an example. Let \( x = 5 \) and \( y = 3 \), so \( x > y \), and let \( z = -2 \), so \( z < 0 \). Then,

\[ \frac{x}{z} = \frac{5}{-2} = -2.5 \quad \text{and} \quad \frac{y}{z} = \frac{3}{-2} = -1.5. \]

Since \( -2.5 < -1.5 \), the inequality \[ \frac{x}{z} < \frac{y}{z} \] is satisfied.