Question 24

a = -1

Given that \( b < 0 \), the expression \( a = \frac{b}{|b|} \) simplifies to \( a = \frac{b}{-b} = -1 \). Therefore, it must be true that \( a = -1 \).

A) a = |b|

This option is incorrect because \( |b| \) represents the absolute value of \( b \), which is positive since \( b < 0 \). Therefore, \( a \) cannot equal \( |b| \), as \( a \) is negative.

B) a = b

This option is incorrect. Since \( b < 0 \) and \( a = -1 \), \( a \) cannot equal \( b \) because \( -1 \) is not equal to any negative value of \( b \).

C) a = -b

This option is incorrect. If \( b < 0 \), then \( -b \) is positive. Since \( a = -1 \), it cannot be equal to a positive number like \( -b \).

D) a = 1

This option is incorrect. Given that \( a = -1 \), it cannot equal \( 1 \) as they are clearly opposites.

E) a = -1

This option is correct. As derived from the expression \( a = \frac{b}{|b|} \) and knowing \( b < 0 \), it leads to \( a = -1 \).

Conclusion

The correct answer is \( a = -1 \) because it directly follows from the given expression when \( b \) is negative. All other options fail as they either lead to contradictions or do not satisfy the conditions set by the question.