1Cademy - Solving $$\frac{1}{x} + \frac{1}{3} = \frac{5}{6}$$

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Strategy to Solve Equations with Rational Expressions

Example

Solving $\frac{1}{x} + \frac{1}{3} = \frac{5}{6}$

Solve the rational equation $\frac{1}{x} + \frac{1}{3} = \frac{5}{6}$ by applying the five-step strategy for equations with rational expressions.

Step 1 — Identify restricted values. The fraction $\frac{1}{x}$ has the variable $x$ in its denominator. If $x = 0$ , this expression is undefined, so record the restriction $x \neq 0$ .

Step 2 — Find the LCD. The denominators in the equation are $x$ , $3$ , and $6$ . The least common denominator of these three is $6x$ .

Step 3 — Clear the fractions. Multiply both sides of the equation by $6x$ and use the Distributive Property:

$6x \cdot \frac{1}{x} + 6x \cdot \frac{1}{3} = 6x \cdot \frac{5}{6}$

Simplify each term: $6x \cdot \frac{1}{x} = 6$ , $6x \cdot \frac{1}{3} = 2x$ , and $6x \cdot \frac{5}{6} = 5x$ . The equation is now fraction-free:

$6 + 2x = 5x$

Step 4 — Solve the resulting equation. Subtract $2x$ from both sides: $6 = 3x$ . Divide both sides by $3$ : $x = 2$ .

Step 5 — Check. The value $x = 2$ does not equal the restricted value $0$ , so it is not extraneous. Substitute $x = 2$ into the original equation:

$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Since $\frac{5}{6} = \frac{5}{6}$ is true, $x = 2$ is confirmed as the solution.

This example demonstrates the complete five-step strategy applied to a rational equation whose variable appears in a denominator. Unlike clearing fractions in a linear equation — where all denominators are constants — here the LCD contains the variable $x$ , making the restriction in Step 1 essential. The process of multiplying by the LCD of $6x$ eliminates all fractions in one step, converting the rational equation into the simple linear equation $6 + 2x = 5x$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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