The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4/(x^2 + 3x - 10) - 1/(x^2 + x - 6) = 3/(x^2 - x - 12)

Key Concepts

Types of Equations

In algebra, equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity holds true for all values of the variable (e.g., 0 = 0), a conditional equation is true for specific values (e.g., x = 2), and an inconsistent equation has no solution (e.g., x + 1 = x). Understanding these classifications is essential for determining the nature of the given equation.

Factoring Polynomials

Factoring polynomials is a critical skill in solving rational equations. It involves expressing a polynomial as a product of its factors, which can simplify the equation and make it easier to solve. For example, the quadratic expressions in the denominators of the given equation can be factored to identify common terms and simplify the overall equation.

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. When solving equations involving rational expressions, it is important to find a common denominator to combine the fractions effectively. Additionally, one must be cautious of restrictions on the variable that may arise from the denominators, as these can affect the solution set.