Problem 46 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{3 x}{x-3}-\frac{x+4}{x+2}$$

Step-by-Step Solution

Answer

\(\frac{2x^{2} + 13x - 12}{(x - 3)(x + 2)}\)

Step 1: Identify the denominators

From the given equation, the denominators are \(x - 3\) and \(x + 2\). In this step, identify the individual denominators to prepare for finding a common denominator.

Step 2: Find the least common denominator

The least common denominator of two fractions is found by multiplying the two denominators together when they don't have any common factors. In this case, the least common denominator is \((x - 3)(x + 2)\).

Step 3: Rewrite the fractions with common denominator

Rewrite each fraction to have the common denominator. Do this by multiplying the numerator and denominator of the first fraction by \(x + 2\) and the numerator and denominator of the second fraction by \(x - 3\). This gives: \(\frac{3x(x + 2)}{(x - 3)(x + 2)} - \frac{(x + 4)(x - 3)}{(x - 3)(x + 2)}\).

Step 4: Simplify the numerators and simplify the common fractions

Simplify the numerators to get: \(\frac{3x^{2} + 6x - x^{2} + 7x - 12}{(x - 3)(x + 2)}\). Simplify this to get:\(\frac{2x^{2} + 13x - 12}{(x - 3)(x + 2)}\).