Questions: Add or subtract the following rational expressions, as indicated, and simplify your answer. (x+4)/(x-2)-(x-3)/(x+6)-4/(x^2+4x-12)

Solution Steps

To solve the problem of adding or subtracting rational expressions, we need to find a common denominator for all the expressions involved. The denominators in this case are \(x-2\), \(x+6\), and \(x^2+4x-12\). First, factor the quadratic expression \(x^2+4x-12\) to find its roots and express it in factored form. Then, determine the least common denominator (LCD) by combining these factors. Rewrite each fraction with the LCD as the new denominator, perform the subtraction, and simplify the resulting expression.

We start with the rational expressions: \[ \frac{x + 4}{x - 2} - \frac{x - 3}{x + 6} - \frac{4}{x^2 + 4x - 12} \]

The quadratic expression \(x^2 + 4x - 12\) can be factored as: \[ x^2 + 4x - 12 = (x - 2)(x + 6) \]

The least common denominator (LCD) for the expressions is: \[ \text{LCD} = (x - 2)(x + 6) \]

Rewriting each expression with the common denominator: \[ \frac{(x + 4)(x + 6)}{(x - 2)(x + 6)} - \frac{(x - 3)(x - 2)}{(x + 6)(x - 2)} - \frac{4}{(x - 2)(x + 6)} \]

Combining the expressions, we have: \[ \frac{(x + 4)(x + 6) - (x - 3)(x - 2) - 4}{(x - 2)(x + 6)} \]

After performing the subtraction and simplifying, we arrive at: \[ \frac{15x + 14}{(x - 2)(x + 6)} \]

Final Answer