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. 1/(x - 1) + 5 = 11/(x - 1)

Verified step by step guidance

1. Identify the denominators in the equation: here, the denominator is \( x - 1 \).
2. Find the values of \( x \) that make the denominator zero by solving \( x - 1 = 0 \). This gives the restriction \( x \neq 1 \) because division by zero is undefined.
3. Rewrite the equation \( \frac{1}{x - 1} + 5 = \frac{11}{x - 1} \) and aim to isolate the variable by eliminating the denominators. Since both fractions have the same denominator, multiply both sides of the equation by \( x - 1 \) to clear the denominators, keeping in mind \( x \neq 1 \).
4. After multiplying, simplify the resulting equation by combining like terms and isolating \( x \). This will give you a linear equation without fractions.
5. Solve the simplified equation for \( x \), then check your solution against the restriction \( x \neq 1 \) to ensure it is valid.

Key Concepts

Rational Expressions and Denominators

A rational expression is a fraction where the numerator and denominator are polynomials. Understanding that the denominator cannot be zero is crucial because division by zero is undefined. Identifying values that make the denominator zero helps determine restrictions on the variable before solving the equation.

Solving Rational Equations

Solving rational equations involves finding values of the variable that satisfy the equation while respecting restrictions. This often requires eliminating denominators by multiplying both sides by the least common denominator (LCD), simplifying, and solving the resulting equation. Always check solutions against restrictions to avoid extraneous roots.

Restrictions on the Variable

Restrictions are values that make any denominator zero and must be excluded from the solution set. Before solving, identify these values to avoid invalid solutions. After solving, verify that none of the solutions violate these restrictions to ensure the final answer is valid.