Question

subtract the rational expressions.

\\\frac{2x - 1}{x^2 + 8x + 12} - \frac{x - 7}{x^2 + 8x + 12}\\

\\\frac{2x - 1}{x^2 + 8x + 12} - \frac{x - 7}{x^2 + 8x + 12} = \text{ } (\text{simplify your answer.})\\

Explanation:

Identify the common denominator

The given problem asks us to subtract two rational expressions:

$$ \frac{2x - 1}{x^2 + 8x + 12} - \frac{x - 7}{x^2 + 8x + 12} $$

Since both fractions already share the same denominator, \(x^2 + 8x + 12\), we can combine their numerators directly over this common denominator.

Subtract the numerators

Subtract the second numerator from the first, being careful to distribute the negative sign to both terms of the second numerator:

$$ \frac{(2x - 1) - (x - 7)}{x^2 + 8x + 12} $$

Distribute the negative sign:

$$ 2x - 1 - x + 7 = x + 6 $$

This gives the combined expression:

$$ \frac{x + 6}{x^2 + 8x + 12} $$

Factor the denominator

To simplify the fraction, we factor the quadratic expression in the denominator, \(x^2 + 8x + 12\). We look for two numbers that multiply to \(12\) and add up to \(8\). These numbers are \(6\) and \(2\):

$$ x^2 + 8x + 12 = (x + 6)(x + 2) $$

Simplify the rational expression

Substitute the factored denominator back into our expression:

$$ \frac{x + 6}{(x + 6)(x + 2)} $$

Since \(x + 6\) is a common factor in both the numerator and the denominator, we can cancel it out (assuming \(x
eq -6\)):

$$ \frac{1}{x + 2} $$

Answer:

Subtract the rational expressions.

$$ \frac{2x-1}{x^2+8x+12} - \frac{x-7}{x^2+8x+12} = $$

<blank>\(\frac{1}{x+2}\)</blank> (Simplify your answer.)