Add or subtract, as indicated. See Example 4. x + y 2x——— - ——— 2x - y y - 2x

Verified step by step guidance

1. Identify a common denominator for the fractions. The denominators are \(2x - y\) and \(y - 2x\). Notice that \(y - 2x = -(2x - y)\).
2. Rewrite the second fraction with the common denominator \(2x - y\) by multiplying the numerator and denominator by -1, resulting in \(-\frac{2x}{2x - y}\).
3. Now, the expression becomes \(\frac{x + y}{2x - y} - \left(-\frac{2x}{2x - y}\right)\).
4. Combine the fractions by adding the numerators: \(\frac{(x + y) + 2x}{2x - y}\).
5. Simplify the numerator by combining like terms: \(x + y + 2x = 3x + y\), resulting in \(\frac{3x + y}{2x - y}\).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions, including addition and subtraction, is crucial for solving problems involving them. This includes finding a common denominator, simplifying the expression, and ensuring that the operations adhere to algebraic rules.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When adding or subtracting rational expressions, it is essential to convert each expression to have this common denominator to perform the operation correctly. This process often involves factoring the denominators and determining the least common multiple.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which can include factoring, canceling common terms, and combining like terms. This step is important after performing operations on rational expressions to ensure the final answer is presented in the most concise and understandable manner. Mastery of simplification techniques is vital for effective problem-solving in algebra and trigonometry.