In Exercises 115–122, find all values of x satisfying the given conditions. y1 = 2x/(x + 2), y2 = 3/(x + 4), and y1 + y2 = 1

Verified step by step guidance

1. Start by setting up the equation based on the given condition: $\frac{2x}{x+2}+\frac{3}{x+4}=1$.
2. To eliminate the fractions, find a common denominator, which is $(x+2)(x+4)$.
3. Multiply every term in the equation by the common denominator $(x+2)(x+4)$ to clear the fractions.
4. Simplify the resulting equation by distributing and combining like terms.
5. Solve the resulting quadratic equation for $x$ by factoring, using the quadratic formula, or completing the square.

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this question, y1 and y2 are rational functions where the numerator and denominator are polynomials. Understanding how to manipulate and combine these functions is essential for solving the equation y1 + y2 = 1.

Finding Common Denominators

To add or equate rational functions, it is often necessary to find a common denominator. This involves identifying the least common multiple of the denominators of the functions involved. In this case, combining y1 and y2 requires finding a common denominator to simplify the equation effectively.

Solving Rational Equations

Solving rational equations involves isolating the variable by eliminating the denominators, often through cross-multiplication or finding a common denominator. Once the equation is simplified, standard algebraic techniques can be applied to solve for x. It is crucial to check for extraneous solutions that may arise from the original rational expressions.