In Exercises 61–66, find all values of x satisfying the given conditions. y1 = (2x - 1)/(x^2 + 2x - 8), y2 = 2/(x + 4), y3 = 1/(x - 2), and y1 + y2 = y3.

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. Understanding their behavior, including asymptotes and discontinuities, is crucial for solving equations involving them. In this question, the functions y1, y2, and y3 are rational, and their properties will influence the solutions for x.

Finding Common Denominators

To solve equations involving rational functions, it is often necessary to find a common denominator. This allows for the combination of fractions and simplifies the equation. In this case, combining y1 and y2 to equal y3 requires identifying a common denominator to facilitate solving for x.

Solving Rational Equations

Solving rational equations involves isolating the variable and eliminating the denominators. This can lead to polynomial equations that can be solved using factoring, the quadratic formula, or other algebraic methods. The equation y1 + y2 = y3 will need to be manipulated to find the values of x that satisfy the equality.