Exercises 57–59 will help you prepare for the material covered in the next section. Subtract: $\frac{3}{x - 4} - \frac{2}{x + 2}$

Verified step by step guidance

1. Identify the two rational expressions to subtract: \(\frac{3}{x-4}\) and \(\frac{2}{x+2}\).
2. Find the least common denominator (LCD) of the two fractions. Since the denominators are \((x-4)\) and \((x+2)\), the LCD is the product of these two distinct factors: \((x-4)(x+2)\).
3. Rewrite each fraction with the LCD as the new denominator by multiplying numerator and denominator appropriately: - For \(\frac{3}{x-4}\), multiply numerator and denominator by \((x+2)\) to get \(\frac{3(x+2)}{(x-4)(x+2)}\). - For \(\frac{2}{x+2}\), multiply numerator and denominator by \((x-4)\) to get \(\frac{2(x-4)}{(x-4)(x+2)}\).
4. Now that both fractions have the same denominator, subtract the numerators: \(\frac{3(x+2) - 2(x-4)}{(x-4)(x+2)}\).
5. Simplify the numerator by distributing and combining like terms, then write the final expression as a single simplified rational expression.

Key Concepts

Finding a Common Denominator

When subtracting rational expressions, you must first find a common denominator. This involves identifying the least common denominator (LCD) that both denominators share, which allows you to rewrite each fraction with the same denominator for easy subtraction.

Subtracting Rational Expressions

After expressing both fractions with a common denominator, subtract the numerators while keeping the denominator the same. This step combines the expressions into a single rational expression, simplifying the subtraction process.

Simplifying the Resulting Expression

Once the subtraction is performed, simplify the resulting rational expression by factoring and reducing common factors in the numerator and denominator. This ensures the expression is in its simplest form for clarity and further use.