If r is a non-zero rational number and x is an irrational number, then the product rx is

Options

A. a rational number
B. an integer
C. an irrational number
D. None of these

Answer

To solve the problem, we need to determine the nature of the product \( rx \) where \( r \) is a non-zero rational number and \( x \) is an irrational number.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- A **rational number** is a number that can be expressed as the quotient of two integers, where the denominator is not zero. For example, \( r = \frac{3}{4} \) or \( r = 2 \).
- An **irrational number** is a number that cannot be expressed as a simple fraction. This means it cannot be written in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers. Examples include \( \sqrt{2} \) and \( \pi \).

**Hint**: Recall the definitions of rational and irrational numbers.

2. **Product of Rational and Irrational**:
- We need to analyze the product \( rx \). Since \( r \) is a non-zero rational number and \( x \) is an irrational number, we can express \( rx \) as \( r \cdot x \).

**Hint**: Remember that multiplying a rational number by an irrational number can change the nature of the result.

3. **Properties of Rational and Irrational Numbers**:
- It is a known property that the product of a non-zero rational number and an irrational number is always irrational. This is because if \( rx \) were rational, then we could express \( x \) as \( \frac{rx}{r} \), which would imply that \( x \) is rational (since \( r \) is non-zero and rational). This contradicts the fact that \( x \) is irrational.

**Hint**: Think about what happens if you assume \( rx \) is rational and try to express \( x \) in terms of \( r \).

4. **Conclusion**:
- Therefore, we conclude that the product \( rx \) is irrational.

**Final Statement**: If \( r \) is a non-zero rational number and \( x \) is an irrational number, then the product \( rx \) is irrational.

### Summary:
The product of a non-zero rational number and an irrational number is always irrational.