In Exercises 61–66, find all values of x satisfying the given conditions. y1 = 5/(x + 4), y2 = 3/(x + 3), y3 = (12x + 19)/(x^2 + 7x + 12). and y1 + y2 = y3.

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this question, y1, y2, and y3 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 = y3.

Finding Common Denominators

To add or equate rational functions, it is often necessary to find a common denominator. This involves identifying a common base for the denominators of the functions involved, which allows for the combination of the fractions into a single expression. This step is crucial for simplifying the equation and solving for x.

Solving Polynomial Equations

Once the rational functions are combined, the resulting equation may lead to a polynomial equation that needs to be solved for x. This involves techniques such as factoring, using the quadratic formula, or applying synthetic division. Mastery of these methods is vital for finding all possible values of x that satisfy the given conditions.