Problem 56 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{x+5}{x^{2}-4}-\frac{x+1}{x-2}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-x^2 -2x + 3}{(x-2)(x+2)}\).

Step 1: Factor the Difference of Squares

The denominator of the first fraction \(x^{2}-4\) is a difference of squares, and thus can be factored as \((x-2)(x+2)\), so the expression becomes \(\frac{x+5}{(x-2)(x+2)} - \frac{x+1}{x-2}\)

Step 2: Make the Denominators the Same

In order to subtract fractions, they need to have the same denominator. Notice that \(x-2\) in the denominator of the second fraction is already a part of the factored form of the denominator of the first fraction. Therefore, multiply the numerator and the denominator of the second expression by \(x+2\), this will make the denominators of both fractions the same. The expression becomes: \(\frac{x+5}{(x-2)(x+2)} - \frac{(x+1)(x+2)}{(x-2)(x+2)}\).

Step 3: Subtract the Fractions

Now that they have the same denominator, you can subtract the numerators: \(\frac{(x+5) - (x+1)(x+2)}{(x-2)(x+2)}\)

Step 4: Simplify the Numerator

Simplify the numerator by distributing and combining like terms: \(\frac{(x+5) - (x^2 + 3x + 2)}{(x-2)(x+2)} = \frac{-x^2 -2x + 3}{(x-2)(x+2)}\)

Step 5: Reduce the Fraction

After simplifying the numerator and denominator, check to see if the fraction can be further reduced. To do this, try to factor the numerator. In this case, the numerator cannot be factored further, thus the fraction is in the simplest form: \(\frac{-x^2 -2x + 3}{(x-2)(x+2)}\).