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. 4x/(x + 3) - 12/(x - 3) = (4x^2 + 36)/(x^2 - 9)

Verified step by step guidance

1. Identify the common denominator for the fractions on both sides of the equation. Notice that $x^2-9$ can be factored as $(x+3)(x-3)$.
2. Multiply every term by the common denominator $(x+3)(x-3)$ to eliminate the fractions.
3. Simplify each term after multiplying by the common denominator. This should clear the fractions and leave you with a polynomial equation.
4. Combine like terms and simplify the resulting polynomial equation.
5. Solve the simplified polynomial equation for $x$ and check the solutions in the original equation to determine if they are valid.

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, a conditional equation is true for specific values, and an inconsistent equation has no solutions. Understanding these classifications is essential for determining the nature of the given equation.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. In the given equation, the presence of rational expressions requires careful manipulation, such as finding a common denominator or simplifying the expressions, to solve the equation effectively. Mastery of operations with rational expressions is crucial for solving complex equations.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, yield the original polynomial. In the context of the given equation, factoring can simplify the rational expressions and help identify potential solutions or restrictions on the variable. This skill is fundamental in solving polynomial equations and analyzing their behavior.