Problem 71 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+3}{x^{2}+x-2}-\frac{2}{x^{2}-1}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x^2 + 2x - 1}{(x-1)(x+2)(x+1)}\).

Step 1: Factor denominators

Factor the denominators \(x^{2}+x-2\) and \(x^{2}-1\). The factored form of \(x^{2}+x-2\) is \((x-1)(x+2)\) and of \(x^{2}-1\) is \((x-1)(x+1)\).

Step 2: Find common denominator

Identify the common denominator for the two expressions. This would be the product of unique factors, which is \((x-1)(x+2)(x+1)\).

Step 3: Rewrite the expression

Rewrite each fraction with the common denominator. Multiply each term by needed factors. The expression becomes \[\frac{(x+3)(x+1)}{(x-1)(x+2)(x+1)} - \frac{2(x+2)}{(x-1)(x+2)(x+1)}\].

Step 4: Simplify

Subtract the fractions, performing the operation in the numerators as the denominator is the same: \[\frac{(x+3)(x+1) - 2(x+2)}{(x-1)(x+2)(x+1)}\].

Step 5: Further simplify the numerator

Expand and simplify the numerator to get \(\frac{x^2 + 2x - 1}{(x-1)(x+2)(x+1)}\).

Step 6: Check if it can be factored

Check to see if the new numerator can be factored. It can't, as there are no factors of -1 that combine to make 2.