Question

Given that a,b and x are real numbers and a< b, x < 0 then

• A. \(\frac{a}{x}<\frac{b}{x}\)
• B. \(\frac{a}{x}>\frac{b}{x}\)
• C. \(\frac{a}{x}≤\frac{b}{x}\)
• D. \(\frac{a}{x}≥\frac{b}{x}\)

Hint:

When dividing an inequality by a negative number, remember that the direction of the inequality reverses. This is an important rule to consider when solving inequalities involving negative divisors.

Answer 1

The Correct Option is B

The correct answer is: (B): \( \frac{a}{x} > \frac{b}{x} \)

We are given that \( a \), \( b \), and \( x \) are real numbers such that \( a < b \) and \( x < 0 \). We are tasked with determining the relationship between \( \frac{a}{x} \) and \( \frac{b}{x} \).

Step 1: Understanding the implications of \( a < b \) and \( x < 0 \)

We are given that \( a < b \), so initially, it would seem that \( \frac{a}{x} \) should be less than \( \frac{b}{x} \). However, since \( x < 0 \), we must consider how dividing by a negative number affects the inequality.

Step 2: Dividing by a negative number

When you divide both sides of an inequality by a negative number, the direction of the inequality reverses. So, dividing \( a < b \) by the negative number \( x \) gives:

\[ \frac{a}{x} > \frac{b}{x} \]

Step 3: Conclusion

Thus, since \( x < 0 \), the inequality reverses, and we have \( \frac{a}{x} > \frac{b}{x} \), which means the correct answer is (B): \( \frac{a}{x} > \frac{b}{x} \).