If a and b are nonzero numbers such that a lt b , which of the following must be true ?
I. `1/a gt 1/b`
II. `a^2 lt b^2`
III.`b^2 ge 1`

Options

A. None
B. I only
C. II only
D. III only

Answer

To solve the problem, we need to analyze each statement given the condition that \( a < b \) where \( a \) and \( b \) are nonzero numbers.

### Step 1: Analyze Statement I: \( \frac{1}{a} > \frac{1}{b} \)

Given \( a < b \), we can consider two cases: when both \( a \) and \( b \) are positive and when both are negative.

- **Case 1: Both \( a \) and \( b \) are positive.**
- If \( a < b \), then \( \frac{1}{a} > \frac{1}{b} \) holds true because the reciprocal of a smaller positive number is larger.

- **Case 2: Both \( a \) and \( b \) are negative.**
- If \( a < b \), then \( a \) is more negative than \( b \) (e.g., \( a = -3 \) and \( b = -1 \)). In this case, \( \frac{1}{a} < \frac{1}{b} \) because the reciprocal of a more negative number is smaller.

Since the statement does not hold true in all cases, **Statement I is not necessarily true.**

### Step 2: Analyze Statement II: \( a^2 < b^2 \)

We can again consider two cases:

- **Case 1: Both \( a \) and \( b \) are positive.**
- If \( a < b \), then squaring both sides gives \( a^2 < b^2 \), which holds true.

- **Case 2: Both \( a \) and \( b \) are negative.**
- If \( a < b \) (e.g., \( a = -3 \) and \( b = -1 \)), squaring gives \( a^2 = 9 \) and \( b^2 = 1 \). Here, \( a^2 > b^2 \).

Since the statement does not hold true in all cases, **Statement II is also not necessarily true.**

### Step 3: Analyze Statement III: \( b^2 \geq 1 \)

This statement depends on the value of \( b \):

- **Case 1: If \( b \) is between 0 and 1.**
- For example, if \( b = \frac{1}{2} \), then \( b^2 = \frac{1}{4} < 1 \).

- **Case 2: If \( b \) is greater than or equal to 1.**
- In this case, \( b^2 \geq 1 \) holds true.

Since \( b \) can take values less than 1, **Statement III is also not necessarily true.**

### Conclusion

None of the statements I, II, or III must be true given that \( a < b \). Therefore, the answer is that none of the statements are necessarily true.

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