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: \(x + 2\), \(x - 2\), and \((x + 2)(x - 2)\). To find the restrictions on the variable, set each denominator equal to zero and solve for \(x\). This gives the values that make the denominators undefined.
2. Set \(x + 2 = 0\) which implies \(x = -2\), and set \(x - 2 = 0\) which implies \(x = 2\). These values are restrictions and cannot be solutions because they make the denominators zero.
3. Rewrite the original equation: \(\frac{3}{x + 2} + \frac{2}{x - 2} = \frac{8}{(x + 2)(x - 2)}\). To solve, multiply every term by the least common denominator (LCD), which is \((x + 2)(x - 2)\), to eliminate the denominators.
4. After multiplying through by the LCD, simplify each term: \$3(x - 2) + 2(x + 2) = 8$. This step removes the fractions and results in a polynomial equation.
5. Expand and combine like terms on the left side, then solve the resulting linear equation for \(x\). Remember to check that your solutions do not include the restricted values \(x = -2\) or \(x = 2\).

Key Concepts

Restrictions on the Variable

In rational equations, values that make any denominator zero are excluded from the solution set because division by zero is undefined. Identifying these restrictions involves setting each denominator equal to zero and solving for the variable. These values must be excluded before solving the equation to avoid invalid solutions.

Solving Rational Equations

To solve rational equations, first find a common denominator to combine terms or clear denominators by multiplying both sides of the equation. This process transforms the equation into a polynomial or simpler form, making it easier to solve. Always check solutions against restrictions to ensure validity.

Factoring and Simplifying Expressions

Factoring expressions, especially denominators like (x + 2)(x - 2), helps identify restrictions and simplifies the equation. Recognizing factored forms allows for easier manipulation and combination of terms. Simplifying expressions correctly is essential to avoid errors in solving rational equations.