Questions: Simplify the following rational expression. (x^2-1)/(x^2+5x-6) Give your answer as a reduced rational expression.

Simplify the following rational expression. (x^2-1)/(x^2+5x-6) Give your answer as a reduced rational expression.
Transcript text: Simplify the following rational expression. \[ \frac{x^{2}-1}{x^{2}+5 x-6} \] Give your answer as a reduced rational expression. Provide your answer below:
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Solution

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Solution Steps

Step 1: Factor the Numerator and Denominator

We start with the rational expression

\[ \frac{x^{2}-1}{x^{2}+5x-6} \]

First, we factor the numerator \(x^{2} - 1\) and the denominator \(x^{2} + 5x - 6\).

The numerator can be factored as:

\[ x^{2} - 1 = (x - 1)(x + 1) \]

The denominator can be factored as:

\[ x^{2} + 5x - 6 = (x - 1)(x + 6) \]

Step 2: Write the Rational Expression in Factored Form

Substituting the factored forms back into the rational expression, we have:

\[ \frac{(x - 1)(x + 1)}{(x - 1)(x + 6)} \]

Step 3: Simplify the Expression

Next, we simplify the expression by canceling the common factor \((x - 1)\) from the numerator and the denominator:

\[ \frac{(x + 1)}{(x + 6)} \]

Final Answer

The simplified rational expression is

\[ \boxed{\frac{x + 1}{x + 6}} \]

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