To simplify the given rational expression \(\frac{x - 6x + 9}{2x - 6}\), let's first simplify the numerator.
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Combine like terms in the numerator: \[ x - 6x + 9 = -5x + 9 \] So, the expression becomes: \[ \frac{-5x + 9}{2x - 6} \]
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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: \[ \frac{-5x + 9}{2(x - 3)} \]
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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 \(-5x + 9\).
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Final simplified form: We do not have any common factors between the numerator and the denominator to simplify further. Thus, the simplified expression is: \[ \frac{-5x + 9}{2(x - 3)} \]
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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: \[ 2x - 6 = 0 \implies 2x = 6 \implies x = 3 \]
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Final Answer: The simplified rational expression is: \[ \frac{-5x + 9}{2(x - 3)} \] The number that must be excluded from the domain is: \[ x = 3 \]
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