Find all values of x satisfying the given conditions. y₁ = (2x - 1)/(x² + 2x - 8), y₂ = 2/(x + 4), y₃ = 1/(x - 2), and y₁ + y₂ = y₃.

Verified step by step guidance

1. Start by substituting the given expressions for y1, y2, and y3 into the equation y1 + y2 = y3. This gives: \( \frac{2x - 1}{x^2 + 2x - 8} + \frac{2}{x + 4} = \frac{1}{x - 2} \).
2. Simplify the denominators where possible. Factorize \( x^2 + 2x - 8 \) into \( (x + 4)(x - 2) \). This changes the equation to: \( \frac{2x - 1}{(x + 4)(x - 2)} + \frac{2}{x + 4} = \frac{1}{x - 2} \).
3. Eliminate the fractions by multiplying through by the least common denominator (LCD), which is \( (x + 4)(x - 2) \). Multiply each term by the LCD to get: \( (2x - 1) + 2(x - 2) = (x + 4) \).
4. Simplify the resulting equation by distributing and combining like terms. Expand \( 2(x - 2) \) to \( 2x - 4 \), and combine terms on the left-hand side: \( 2x - 1 + 2x - 4 = x + 4 \).
5. Solve the simplified linear equation for \( x \). Combine like terms on both sides and isolate \( x \) to find the solution. Be sure to check for any restrictions on \( x \) (e.g., values that make the original denominators undefined).

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 combine rational expressions, finding a common denominator is essential. This process allows for the addition or subtraction of fractions, which is necessary when setting y1 + y2 equal to y3. Mastery of this concept is vital for simplifying the equation and solving for x.

Solving Rational Equations

Solving rational equations involves isolating the variable and eliminating denominators, often by multiplying through by the least common denominator. This step is crucial to avoid undefined values and to find valid solutions for x. Understanding how to manipulate and solve these equations is key to answering the given problem.