Classwork (Simplify \[ \frac{1}{x-4}+\frac{2}{x+3}-\frac{2}{x^{2}-x-12} \] \[ 2 \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}+x-2} \]

To simplify the expression, start by factoring the denominators: The first denominator, \(x^2 - x - 12\), can be factored to \((x - 4)(x + 3)\). The second denominator, \(x^2 + x - 2\), factors to \((x - 1)(x + 2)\). Using these factorizations, rewrite the expression: \[ \frac{1}{x-4} + \frac{2}{x+3} - \frac{2}{(x-4)(x+3)} + 2 \frac{x}{x+2} - \frac{2}{x-1} - \frac{6}{(x-1)(x+2)} \] Next, find a common denominator, which would be the product of all distinct linear factors: \((x-4)(x+3)(x-1)(x+2)\). Combine the fractions by ensuring each has this common denominator, and simplify the numerators accordingly. This leads to a combined expression you can simplify further and solve for specific \(x\) values if desired. Remember to check for restrictions where \(x\) would make any denominator equal to zero! ----- Understanding rational expressions isn’t just about math on paper; it's crucial in fields like engineering and physics! For example, engineers might work with forces acting on varying structures where such expressions help in calculating loads. Recognizing how terms react and combine gives insights into system behavior, making the math not just an abstract concept but a tool for real-world solutions! For anyone looking to dive deeper into this subject, consider studying algebraic fractions and their applications in calculus. A good resource would be "College Algebra" by Michael Sullivan. This textbook breaks down rational expressions, discusses their properties, and shows how they’re used in problem-solving, making advanced topics more approachable and fun!