Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 65a

Chapter 1, Problem 65a

Add or subtract, as indicated. (x + y)/(2x - y) - 2x/(y - 2x)

Verified step by step guidance

Identify the two rational expressions to be combined: $\frac{x + y}{2x - y}$ and $\frac{2x}{y - 2x}$.

Notice that the denominators $2x - y$ and $y - 2x$ are very similar but not the same. Rewrite the second denominator to see the relationship: $y - 2x = -(2x - y)$.

Rewrite the second fraction using this relationship: $\frac{2x}{y - 2x} = \frac{2x}{-(2x - y)} = -\frac{2x}{2x - y}$.

Now the expression becomes $\frac{x + y}{2x - y} - \left(-\frac{2x}{2x - y}\right)$, which simplifies to $\frac{x + y}{2x - y} + \frac{2x}{2x - y}$ because subtracting a negative is addition.

Since both fractions have the same denominator $2x - y$, combine the numerators over the common denominator: $\frac{(x + y) + 2x}{2x - y}$. Then simplify the numerator by combining like terms.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Finding a Common Denominator

To add or subtract rational expressions, you must first find a common denominator. This involves identifying the least common denominator (LCD) that both denominators can divide into, allowing the expressions to be combined into a single fraction.

Simplifying Rational Expressions

Simplifying rational expressions involves factoring numerators and denominators and reducing common factors. This step is crucial after combining fractions to express the result in its simplest form.

Handling Negative Signs and Equivalent Denominators

Recognizing that denominators like (2x - y) and (y - 2x) are negatives of each other helps in rewriting expressions for easier addition or subtraction. Properly managing negative signs ensures accurate combination of terms.

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