Problem 55 - College Algebra

Question

Add or subtract as indicated. $$\frac{x+3}{x^{2}-1}-\frac{x+2}{x-1}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{-x^{2}+x+3}{(x-1)(x+1)}\).

Step 1: Simplify the denominators

The denominator \(x^{2}-1\) can be factored into \((x-1)(x+1)\). The expression then transforms into \(\frac{x+3}{(x-1)(x+1)}-\frac{x+2}{x-1}\)

Step 2: Find a common denominator

The common denominator between \((x-1)(x+1)\) and \(x-1\) is \((x-1)(x+1)\). Therefore, adjust the second fraction to have this common denominator by multiplying its numerator and denominator by \(x+1\). The expression then transforms into \(\frac{x+3}{(x-1)(x+1)}-\frac{(x+2)(x+1)}{(x-1)(x+1)}\)

Step 3: Combine the fractions

The fractions can now be combined to form a single fraction: \(\frac{x+3-(x+2)(x+1)}{(x-1)(x+1)}\).

Step 4: Simplify the fraction

The numerator \(x+3-(x+2)(x+1)\) can be simplified to give \(-x^{2}+x+3\). Therefore, the fraction becomes \(\frac{-x^{2}+x+3}{(x-1)(x+1)}\).