Problem 53 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 x}{x^{2}+3 x-10}-\frac{2 x}{x^{2}+x-6}$$

Step-by-Step Solution

Answer

The short answer after simplifying is \(\frac{x^{2} - x}{(x - 2)(x + 5)(x + 3)}\)

Step 1: Factorize the Denominators

Factorize the denominators of the two fractions:The factors of \(x^{2}+3 x-10\) are \((x - 2)(x + 5)\) and the factors of \(x^{2}+x-6\) are \((x - 2)(x + 3)\)

Step 2: Find the Common Denominator

The common denominator of the two fractions is \((x - 2)(x + 5)(x + 3)\) because it includes all the unique factors of the two denominators.

Step 3: Rewrite the Fractions with the Common Denominator

Rewrite the two fractions using the common denominator:\[\frac{3 x(x + 3)}{(x - 2)(x + 5)(x + 3)} - \frac{2 x(x + 5)}{(x - 2)(x + 5)(x + 3)}\]

Step 4: Subtract the Numerators

Subtract the numerators and keep the denominator the same:\[\frac{3 x(x + 3) - 2 x(x + 5)}{(x - 2)(x + 5)(x + 3)}\]

Step 5: Simplify the Numerator

Simplify the numerator by opening the brackets:\[\frac{3 x^{2} + 9 x - 2 x^{2} - 10 x}{(x - 2)(x + 5)(x + 3)}\] becomes \[\frac{x^{2} - x}{(x - 2)(x + 5)(x + 3)}\]