Problem 56 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Factorize the Denominator x² - 4

First, factorize the denominator of the first term. The expression \(x^{2}-4\) is a difference of two squares, which can be factorized as \((x-2)(x+2)\). Rewrite the first fraction as \(\frac{x+5}{(x-2)(x+2)}\). Hence, the whole expression becomes \(\frac{x+5}{(x-2)(x+2)}-\frac{x+1}{x-2}\).

Step 2: Obtain Common Denominator

To combine the fractions, common denominator is required. The least common denominator (LCD) of the fractions \(\frac{x+5}{(x-2)(x+2)}\) and \(\frac{x+1}{x-2}\) is \((x-2)(x+2)\). Hence, rewrite \(\frac{x+1}{x-2}\) with denominator \((x-2)(x+2)\) which makes it \(\frac{(x+1)(x+2)}{(x-2)(x+2)}\) since multiplying by 1 in the form of \(\frac{x+2}{x+2}\) does not change the fraction.

Step 3: Combining the Fractions

Subtract the two expressions: \(\frac{x+5}{(x-2)(x+2)}-\frac{(x+1)(x+2)}{(x-2)(x+2)}\), which will transform into a single rational expression with the common denominator.

Step 4: Simplifying Numerator

Combine the numerators in the expression: \(\frac{x+5-(x+1)(x+2)}{(x-2)(x+2)}\). Therefore, distribute the negative sign through the expression \(-(x+1)(x+2)\), resulting in the expression \(\frac{x+5-x^{2}-3x-2}{(x-2)(x+2)}\). Combine like terms to get the numerator in standard form: \(-x^{2}-2x+3\).

Step 5: Write the Final Simplified Expression

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