Problem 57 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-5}{x+3}+\frac{x+3}{x-5}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{x - 1}{x^2 -1}\)

Step 1: Identify the common denominator

To add or subtract fractions, we need a common denominator. Here both denominators \(x + 3\) and \(x - 5\) are different, which makes them the least common denominator (LCD).

Step 2: Express the fractions with the common denominator

Express each of the fractions with the common denominator. Since in the denominator, \(x + 3\) and \(x - 5\) are the same for both fractions, we multiply the first fraction by \(\frac{x-5}{x-5}\) and the second fraction by \(\frac{x+3}{x+3}\).\n This gives us \(\frac{x-5}{x^2 -2}+\frac{x+3}{x^2 -2}\).

Step 3: Add the fractions

Since the denominators are the same, we can add the numerators: \(\frac{x - 5 + x +3}{x^2 - 2}\) simplifying the numerator, we get \(\frac{2x - 2}{x^2 -2}\)

Step 4: Simplify the result

Finally, we simplify the fraction by noticing a common factor 2 in both numerator and denominator. Dividing both parts by 2, we get \( \frac{x - 1}{x^2 -1}\). Notice that the denominator can be further factorized, but it's not necessary, since we can't cancel any terms from the numerator and denominator. The expression is simplified to its lowest terms.