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) + 3/(x + 5) = 7/(x + 5)(x - 2)

Verified step by step guidance

1. Identify the least common denominator (LCD) for the fractions, which is \((x - 2)(x + 5)\).
2. Multiply every term in the equation by the LCD to eliminate the fractions: \((x - 2)(x + 5)\left(\frac{4}{x - 2}\right) + (x - 2)(x + 5)\left(\frac{3}{x + 5}\right) = (x - 2)(x + 5)\left(\frac{7}{(x + 5)(x - 2)}\right)\).
3. Simplify each term: \(4(x + 5) + 3(x - 2) = 7\).
4. Expand and combine like terms: \(4x + 20 + 3x - 6 = 7\).
5. Combine the terms to form a linear equation: \(7x + 14 = 7\), and solve for \(x\).

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 that involve polynomials in the numerator and denominator. In the given equation, the terms involve rational expressions, which require careful manipulation, such as finding a common denominator, to solve. Mastery of operations with rational expressions is crucial for simplifying and solving the equation.

Solving Rational Equations

To solve rational equations, one typically eliminates the denominators by multiplying through by the least common denominator (LCD). This process simplifies the equation, allowing for easier manipulation and solution. It is important to check for extraneous solutions that may arise from multiplying by expressions that could be zero.