If a = b/|b| and b < 0, which of the following must be true?

Rationale

a = -1.

Given that ( b < 0 ), the expression ( |b| ) equals ( -b ). Thus, substituting into the equation ( a = b\|b| ) yields ( a = b\-b = -1 ). Therefore, ( a) must equal (-1).

A (a = |b|) YOUR ANSWER

This choice is incorrect because ( |b| ) represents the positive magnitude of ( b ), which is a positive number since ( b < 0 ). Thus, ( a ) cannot equal the positive value of ( |b| ), as ( a ) is determined to be (-1).

B (a = b) YOUR ANSWER

This option is false since ( b ) is negative and ( a ) has been determined to be (-1). Since ( b ) is less than zero, it cannot equal (-1) unless ( b ) specifically equals (-1), which isn't guaranteed. Therefore, this is not a universally true statement.

C (a = -b) YOUR ANSWER

This choice is incorrect because ( -b ) is a positive number (since ( b < 0 )), while ( a = -1 ) is negative. Therefore, ( a ) cannot equal (-b).

D (a = 1) YOUR ANSWER

This statement is incorrect as it contradicts our finding that ( a = -1 ). Since ( a ) has been determined to be negative, it cannot equal the positive value of ( 1).

E (a = -1) CORRECT

This choice is correct, as derived from the equation ( a = b\|b|). Since ( |b| = -b ) for ( b < 0 \), substituting leads to ( a = b\-b = -1).

Conclusion

In the context of the problem where ( a = b\|b| ) and ( b < 0 ), the only statement that holds true is ( a = -1 ). All other choices either contradict this conclusion or misinterpret the relationships between ( a ) and ( b ). This reinforces the importance of understanding absolute values and their implications in inequality contexts.

If a = b/|b| and b < 0, which of the following must be true?

a = -1.

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