Add or subtract, as indicated. See Example 4. (3x)/(x² + x − 12) − x/(x² − 16) + x

Verified step by step guidance

1. First, identify the two rational expressions you need to subtract: \(\frac{3x}{x^{2} + x - 12}\) and \(\frac{x}{x^{2} - 16}\).
2. Next, factor the denominators to find their factored forms: factor \(x^{2} + x - 12\) and \(x^{2} - 16\).
3. After factoring, determine the least common denominator (LCD) by combining all unique factors from both denominators.
4. Rewrite each fraction with the LCD as the new denominator by multiplying numerator and denominator by the necessary factors to match the LCD.
5. Finally, subtract the numerators over the common denominator and simplify the resulting expression if possible.

Key Concepts

Factoring Quadratic Expressions

Factoring involves rewriting quadratic expressions as products of binomials. This is essential for simplifying rational expressions by identifying common factors in numerators and denominators, which helps in performing addition or subtraction of fractions.

Finding a Common Denominator

To add or subtract rational expressions, you must find a common denominator, typically the least common multiple (LCM) of the denominators. This allows the expressions to be combined into a single fraction with a unified denominator.

Adding and Subtracting Rational Expressions

Once the denominators are the same, add or subtract the numerators directly while keeping the common denominator. Simplify the resulting expression by combining like terms and factoring if possible.