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. Identify the denominators in the equation: $x-2$.
2. Determine the value that makes the denominator zero: Set $x-2=0$ and solve for $x$.
3. The restriction is $x\ne 2$ because it makes the denominator zero.
4. To solve the equation, multiply every term by $x-2$ to eliminate the denominators.
5. Simplify the resulting equation and solve for $x$, keeping in mind the restriction $x\ne 2$.

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 identify any restrictions on the variable, particularly values that would make the denominator zero, as these values are not permissible in the solution set.

Restrictions on Variables

Restrictions on variables in rational equations arise when the denominator equals zero, leading to undefined expressions. Identifying these restrictions is crucial because they determine the valid values for the variable, ensuring that the solutions do not include any values that would invalidate the equation.

Solving Rational Equations

To solve rational equations, one typically eliminates the denominators by multiplying both sides of the equation by the least common denominator (LCD). After simplifying, the resulting equation can be solved for the variable, while always considering the previously identified restrictions to ensure the solutions are valid.