Answer
Matt saw that the denominator of the first fraction was the same as the numerator of the second fraction, so he tried to cross-cancel to get rid of them. This is not a situation in which you can cross-cancel. You can only cross cancel when the two rational expressions are multiplied, not when they are added together.
$\frac{2x^2 - 9x + 53}{(x - 7)(x + 4)}$
Work Step by Step
$\frac{x + 1}{x - 7} + \frac{x - 7}{x + 4}$
The following is the correct way to evaluate this expression:
To add or subtract rational expressions, we need to make sure that the expressions have the same denominator.
We need to rewrite the two expressions with the same denominator, so we need to find the least common denominator (LCD) for both expressions.
The first thing we want to do is to make sure that each denominator is factored completely. This has already been done.
Next, we want to find the least common denominator (LCD). We do this by taking the highest power of each factor in the denominator:
LCD = $(x - 7)(x + 4)$
Now that we have the least common denominator, we multiply the numerator of each fraction with the factor or factors it is missing in its denominator:
$\frac{(x + 1)(x + 4)}{(x - 7)(x + 4)} + \frac{(x - 7)(x - 7)}{(x - 7)(x + 4)}$
Rewrite the two fractions as one with the same denominator:
$\frac{(x + 1)(x + 4) + (x - 7)(x - 7)}{(x - 7)(x + 4)}$
Use the distributive property to rewrite the numerator:
$\frac{(x^2 + 5x + 4) + (x^2 - 14x + 49)}{(x - 7)(x + 4)}$
Combine like terms in the numerator:
$\frac{2x^2 - 9x + 53}{(x - 7)(x + 4)}$
