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.

Verified step by step guidance

1. Start by setting up the equation based on the given condition: \( y_1 + y_2 = y_3 \). Substitute the expressions for \( y_1, y_2, \) and \( y_3 \) into this equation: \( \frac{2x - 1}{x^2 + 2x - 8} + \frac{2}{x + 4} = \frac{1}{x - 2} \).
2. To solve this equation, find a common denominator for the fractions on the left side. The denominators are \( x^2 + 2x - 8 \) and \( x + 4 \). Factor \( x^2 + 2x - 8 \) to get \( (x + 4)(x - 2) \). Thus, the common denominator is \( (x + 4)(x - 2) \).
3. Rewrite each fraction with the common denominator: \( \frac{2x - 1}{(x + 4)(x - 2)} + \frac{2(x - 2)}{(x + 4)(x - 2)} = \frac{1}{x - 2} \).
4. Combine the fractions on the left side: \( \frac{(2x - 1) + 2(x - 2)}{(x + 4)(x - 2)} = \frac{1}{x - 2} \). Simplify the numerator: \( 2x - 1 + 2x - 4 = 4x - 5 \).
5. Now, equate the numerators since the denominators are the same: \( 4x - 5 = 1 \). Solve for \( x \) by isolating \( x \) on one side of the equation.

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.