Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 3/(x + 2) + 2/(x - 2) = 8/(x + 2)(x - 2)

Verified step by step guidance

1. Identify the denominators in the equation: $x+2$ and $x-2$.
2. Determine the values of $x$ that make each denominator zero: $x+2=0$ gives $x=-2$ and $x-2=0$ gives $x=2$. These are the restrictions.
3. Multiply every term by the least common denominator $(x+2)(x-2)$ to eliminate the fractions.
4. Simplify the resulting equation by distributing and combining like terms.
5. Solve the simplified equation for $x$, keeping in mind the restrictions $x\ne -2$ and $x\ne 2$.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator and denominator. To solve these equations, it is essential to find a common denominator and eliminate the fractions, which simplifies the equation. Understanding how to manipulate these fractions is crucial for solving rational equations effectively.

Restrictions on Variables

Restrictions on variables in rational equations arise when the denominator equals zero, as division by zero is undefined. Identifying these restrictions involves setting each denominator to zero and solving for the variable. These restrictions must be considered when solving the equation to ensure that the solutions do not lead to undefined expressions.

Solving Rational Equations

To solve rational equations, one typically clears the fractions by multiplying both sides by the least common denominator (LCD). After eliminating the denominators, the resulting equation can be solved using algebraic techniques. It is important to check the solutions against the original restrictions to confirm they are valid.