Solve each equation. See Example 1. x/(x-3) = 3/(x-3) + 3

Verified step by step guidance

1. Start by examining the equation: \(\frac{x}{x-3} = \frac{3}{x-3} + 3\). Notice that the denominators on the left and right sides are the same, which suggests we can work towards eliminating the fractions.
2. To eliminate the denominators, multiply both sides of the equation by the common denominator \((x-3)\), making sure to apply the distributive property correctly: \((x-3) \times \frac{x}{x-3} = (x-3) \times \frac{3}{x-3} + (x-3) \times 3\).
3. Simplify each term after multiplication: the \((x-3)\) in the denominators will cancel with the multiplication, leaving you with \(x = 3 + 3(x-3)\).
4. Next, distribute the 3 on the right side: \(x = 3 + 3x - 9\).
5. Combine like terms on the right side and then isolate \(x\) by moving all \(x\) terms to one side and constants to the other. This will give you a linear equation to solve for \(x\).

Key Concepts

Solving Rational Equations

Rational equations involve expressions with variables in the denominator. To solve them, identify restrictions on the variable to avoid division by zero, then eliminate denominators by multiplying both sides by the least common denominator (LCD). This simplifies the equation to a polynomial form that can be solved using algebraic methods.

Domain Restrictions

Domain restrictions are values that make any denominator zero, which are not allowed in the solution set. Before solving, determine these values to exclude them from possible solutions. After solving, check each solution against these restrictions to ensure it is valid.

Simplifying and Solving Linear Equations

Once denominators are cleared, the resulting equation is often linear. Use standard algebraic techniques such as combining like terms, isolating the variable, and performing inverse operations to find the solution. Verify the solution by substituting it back into the original equation.