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. 2/x + 1/2 = 3/4

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., 2x = 2x), a conditional equation is true for specific values of the variable (e.g., x + 2 = 5), 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.

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. To solve these equations, one typically finds a common denominator to eliminate the fractions, allowing for easier manipulation and simplification. This process often involves cross-multiplication and careful handling of any restrictions on the variable to avoid division by zero.

Checking Solutions

After solving an equation, it is crucial to check the solution by substituting it back into the original equation. This verification step ensures that the solution is valid and helps identify the type of equation. For instance, if the left-hand side equals the right-hand side after substitution, the equation is confirmed as either an identity or a conditional equation, while a mismatch indicates an inconsistency.