Question

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression in order for it to be equivalent to the original expression.

x -- 6x+9
2x-6

Simplify the rational expression. Select the correct choice below and fill in any answer boxes in your choice.

x-6x+9
2x-6
(Simplify your answer. Use positive exponents only. Use integers or fractions for any numbers in the expression.)

Answer 1

To simplify the given rational expression \(\frac{x - 6x + 9}{2x - 6}\), let's first simplify the numerator.

1. **Combine like terms in the numerator:**
\[
x - 6x + 9 = -5x + 9
\]
So, the expression becomes:
\[
\frac{-5x + 9}{2x - 6}
\]

2. **Factor what can be factored:**
In the denominator, \(2x - 6\) can be factored out:
\[
2x - 6 = 2(x - 3)
\]

Thus, the expression now looks like:
\[
\frac{-5x + 9}{2(x - 3)}
\]

3. **Factor the numerator if possible:**
We can rearrange the numerator:
\[
-5x + 9 = -5(x - \frac{9}{5})
\]

However, this does not lead to further simplification with respect to canceling terms. Therefore, we will just leave it as \(-5x + 9\).

4. **Final simplified form:**
We do not have any common factors between the numerator and the denominator to simplify further. Thus, the simplified expression is:
\[
\frac{-5x + 9}{2(x - 3)}
\]

5. **Identify the restrictions on the domain:**
The original expression has a denominator of \(2x - 6\) that must not equal 0. So, we set the denominator equal to zero to find the excluded value:
\[
2x - 6 = 0 \implies 2x = 6 \implies x = 3
\]

6. **Final Answer:**
The simplified rational expression is:
\[
\frac{-5x + 9}{2(x - 3)}
\]
The number that must be excluded from the domain is:
\[
x = 3
\]