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. 5/2x - 8/9 = 1/18 - 1/3x

Verified step by step guidance

1. Identify the denominators in the equation \(\frac{5}{2x} - \frac{8}{9} = \frac{1}{18} - \frac{1}{3x}\). The denominators are \$2x\(, \(9\), \(18\), and \)3x$.
2. Find the values of \(x\) that make any denominator zero. Set each denominator equal to zero: \$2x = 0\( and \)3x = 0\(. Solve these to find the restrictions on \)x$.
3. Rewrite the equation to have a common denominator or multiply both sides by the least common denominator (LCD) to eliminate the fractions. The LCD here is the least common multiple of \$2x\(, \(9\), \(18\), and \)3x$.
4. After clearing the denominators, simplify the resulting equation by combining like terms and isolating the variable \(x\) on one side.
5. Solve the simplified equation for \(x\), then check your solution(s) against the restrictions found in step 2 to ensure no solution makes a denominator zero.

Key Concepts

Rational Equations and Denominators

Rational equations involve expressions with variables in the denominator. Understanding how to handle these is crucial because denominators cannot be zero, as division by zero is undefined. Identifying values that make denominators zero helps determine restrictions on the variable before solving.

Finding Restrictions on the Variable

Restrictions are values that make any denominator in the equation zero. To find them, set each denominator equal to zero and solve for the variable. These values must be excluded from the solution set to avoid undefined expressions.

Solving Rational Equations

To solve rational equations, first find a common denominator to eliminate fractions by multiplying both sides. Then solve the resulting equation, keeping in mind the restrictions. Finally, check solutions against restrictions to ensure no invalid values are included.