Is it possible for rational numbers `(a)/(b) and (b)/(a)`, both in standard form, to be equal, given that `a neb`? Justify your answer with examples.

Answer

To determine if it is possible for the rational numbers \( \frac{a}{b} \) and \( \frac{b}{a} \) to be equal, given that \( a \neq b \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Rational Numbers**:
- A rational number is defined as a number that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

2. **Setting Up the Equation**:
- We need to check if \( \frac{a}{b} = \frac{b}{a} \) can hold true under the condition that \( a \neq b \).

3. **Cross-Multiplying**:
- To eliminate the fractions, we can cross-multiply:
\[ a \cdot a = b \cdot b \]
This simplifies to:
\[ a^2 = b^2 \]

4. **Analyzing the Result**:
- The equation \( a^2 = b^2 \) implies that \( a = b \) or \( a = -b \). However, since we have the condition that \( a \neq b \), the only possibility left is \( a = -b \).

5. **Conclusion**:
- If \( a = -b \), then \( \frac{a}{b} = \frac{-b}{b} = -1 \) and \( \frac{b}{a} = \frac{b}{-b} = -1 \). In this case, both rational numbers are equal, but this contradicts the condition that \( a \neq b \) unless we consider the case where one is the negative of the other.

6. **Example**:
- Let's take \( a = 2 \) and \( b = 3 \):
\[ \frac{a}{b} = \frac{2}{3} \quad \text{and} \quad \frac{b}{a} = \frac{3}{2} \]
Clearly, \( \frac{2}{3} \neq \frac{3}{2} \).

### Final Justification:
- Therefore, it is not possible for \( \frac{a}{b} \) and \( \frac{b}{a} \) to be equal if \( a \) and \( b \) are distinct rational numbers, except in the case where one is the negative of the other.