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. 2/(x - 2) = x/(x - 2) - 2

Verified step by step guidance

1. Step 1: Identify the denominators in the equation. The denominators are (x - 2) and (x - 2).
2. Step 2: Determine the restrictions on the variable by setting each denominator equal to zero. Solve x - 2 = 0, which gives x = 2. This means x = 2 is a restriction, as it would make the denominator undefined.
3. Step 3: Multiply through the entire equation by the least common denominator (LCD), which is (x - 2), to eliminate the fractions. Be cautious not to multiply by (x - 2) if x = 2, as this is a restricted value.
4. Step 4: After clearing the fractions, simplify the resulting equation. You will have a linear or quadratic equation to solve.
5. Step 5: Solve the simplified equation for x, and check your solution(s) against the restriction x = 2. Discard any solution that violates the restriction.

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 identify common denominators and simplify the expressions. Understanding how to manipulate these fractions is crucial for finding solutions while ensuring that the denominators do not equal zero.

Restrictions on Variables

Restrictions on variables arise when the denominator of a rational expression is set to zero, as division by zero is undefined. To find these restrictions, one must solve the equation formed by setting the denominator equal to zero. Identifying these values is critical because they indicate which values cannot be included in the solution set of the equation.

Solving Equations

Solving equations involves finding the values of the variable that satisfy the equation. In the context of rational equations, this often requires isolating the variable and simplifying the equation after addressing any restrictions. It is important to check the solutions against the restrictions to ensure they are valid and do not result in undefined expressions.