Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Rational Equations

College Algebra

1. Equations & Inequalities

Rational Equations

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1:47 minutes

Problem 79

Textbook Question

Solve each equation. x/x+2 + 1/x+3 = 2/x²+2x

Verified step by step guidance

Identify the least common denominator (LCD) for the fractions, which is \( (x+2)(x+3) \).

Multiply each term in the equation by the LCD to eliminate the fractions: \( (x+2)(x+3) \cdot \frac{x}{x+2} + (x+2)(x+3) \cdot \frac{1}{x+3} = (x+2)(x+3) \cdot \frac{2}{x^2+2x} \).

Simplify each term: \( x(x+3) + (x+2) = 2 \).

Combine like terms and simplify the equation: \( x^2 + 3x + x + 2 = 2 \).

Rearrange the equation to form a quadratic equation: \( x^2 + 4x + 2 = 2 \).

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Key Concepts

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

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including finding a common denominator and simplifying, is crucial for solving equations involving them. In this question, the rational expressions must be combined and simplified to isolate the variable.

Finding a Common Denominator

To solve equations involving rational expressions, it is often necessary to find a common denominator. This allows for the combination of fractions into a single expression, making it easier to solve for the variable. In the given equation, identifying the least common denominator will help eliminate the fractions and simplify the solving process.

Cross-Multiplication

Cross-multiplication is a technique used to solve equations involving two fractions set equal to each other. By multiplying the numerator of one fraction by the denominator of the other, we can eliminate the fractions and create a polynomial equation. This method is particularly useful in this problem to simplify the equation and isolate the variable.