Certainly! Let's go through the problem step-by-step.
We are given:
1. [tex]\( x \)[/tex] is a rational number.
2. [tex]\( y = 2 \)[/tex] is an irrational number.
We need to determine the type of number that results from multiplying [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Definitions:
- A rational number is any number that can be expressed as the quotient or fraction [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex].
- An irrational number is a number that cannot be expressed as a simple fraction. It cannot be written as the quotient of two integers.
### Concept:
When a rational number is multiplied by an irrational number, the result is always an irrational number. This is because the product of a non-zero rational number and an irrational number cannot be expressed as a simple fraction, maintaining the characteristics of an irrational number.
### Solution:
1. Let [tex]\( x \)[/tex] be a rational number.
2. Let [tex]\( y = 2 \)[/tex], an irrational number.
3. The product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] will be [tex]\( x \cdot y \)[/tex].
Since [tex]\( y \)[/tex] is irrational and [tex]\( x \)[/tex] is rational, their product [tex]\( x \cdot y \)[/tex] will be irrational.
Conclusion: The type of number that [tex]\( x \cdot y \)[/tex] (which is [tex]\( x \times 2 \)[/tex]) results in is an irrational number.
Therefore, the result from multiplying a rational number [tex]\( x \)[/tex] by the irrational number [tex]\( 2 \)[/tex] is an irrational number.
