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. 7/2x - 5/3x = 22/3

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1. Identify the denominators in the equation \(\frac{7}{2x} - \frac{5}{3x} = \frac{22}{3}\). The denominators are \$2x\( and \)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 on the left side. The common denominator for \$2x\( and \)3x\( is \)6x\(. Express each fraction with denominator \)6x$:
4. \(\frac{7}{2x} = \frac{7 \times 3}{6x} = \frac{21}{6x}\) and \(\frac{5}{3x} = \frac{5 \times 2}{6x} = \frac{10}{6x}\).
5. Combine the fractions on the left side: \(\frac{21}{6x} - \frac{10}{6x} = \frac{21 - 10}{6x} = \frac{11}{6x}\). So the equation becomes \(\frac{11}{6x} = \frac{22}{3}\).
6. Solve for \(x\) by cross-multiplying: \(11 \times 3 = 22 \times 6x\). Simplify and solve the resulting equation for \(x\), keeping in mind the restrictions found earlier.

Key Concepts

Rational Expressions and Denominators

Rational expressions are fractions that contain variables in the denominator. Understanding how to identify values that make denominators zero is crucial because these values are undefined and must be excluded from the solution set to avoid division by zero.

Finding Restrictions on the Variable

Restrictions are values of the variable that cause any denominator in the equation to be zero. Before solving, determine these values to ensure they are not included in the final solution, maintaining the equation's validity.

Solving Rational Equations

To solve rational equations, first clear denominators by multiplying both sides by the least common denominator (LCD). Then solve the resulting equation, and finally check solutions against restrictions to exclude any invalid answers.