Is $b

Question

Is $b < a$?

Given the statements:

(a) $b = \frac{1}{a} - 5$

(b) $a < 0$

(c) $|b| > |a|$

Determine which of the following options is correct regarding the sufficiency of the statements to answer the question.

Accepted Answer

**Solution:** 1. From statement (a): $$b = \frac{1}{a} - 5$$ This can be rewritten as: $$b = \frac{1 - 5a}{a}$$ The sign of $b$ depends on $a$. If $a$ is positive, then $b$ is positive and since $\frac{1}{a}$ is positive, $b$ could be greater than $a$. If $a$ is negative, $b$ could be less than $a$. Hence, statement (a) alone does not provide a definitive answer. 2. From statement (b): $$a < 0$$ This tells us $a$ is negative but gives no information about $b$ directly, so it is insufficient alone. 3. From statement (c): $$|b| > |a|$$ This means the absolute value of $b$ is greater than that of $a$. Consider the cases: - If $b$ is positive and $a$ is negative, then $b > a$. - If $a$ is positive and $b$ is negative, then $a > b$. - If both are positive, then $b > a$. - If both are negative, then $b < a$. So, statement (c) alone is not sufficient. 4. Combining (a) and (b): Since $a < 0$ and $b = \frac{1}{a} - 5$, with $a$ negative, $\frac{1}{a}$ is also negative and subtracting 5 makes $b$ even more negative. Hence, $b < a$. Therefore, statements (a) and (b) together are sufficient. 5. Combining (b) and (c): Knowing $a < 0$ and $|b| > |a|$ does not clarify the sign of $b$, so insufficient. 6. Combining (a) and (c): Without knowing the sign of $a$, this combination is also insufficient. **Answer:** Only statements (a) and (b) together are sufficient to answer the question.

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