1Cademy - Solving $$\frac{5}{3u-2} = \frac{3}{2u}$$

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Strategy to Solve Equations with Rational Expressions

Example

Solving $\frac{5}{3u-2} = \frac{3}{2u}$

Solve the rational equation $\frac{5}{3u - 2} = \frac{3}{2u}$ using the five-step strategy for equations with rational expressions. This example features two distinct variable expressions in the denominators.

Step 1 — Identify restricted values. Setting each denominator equal to zero: $3u - 2 = 0$ gives $u = \frac{2}{3}$ , and $2u = 0$ gives $u = 0$ . Record $u \neq \frac{2}{3}$ and $u \neq 0$ .

Step 2 — Find the LCD. The denominators $(3u - 2)$ and $2u$ share no common factors, so the LCD is their product: $2u(3u - 2)$ .

Step 3 — Clear the fractions. Multiply both sides by the LCD $2u(3u - 2)$ :

$2u(3u - 2) \cdot \frac{5}{3u - 2} = 2u(3u - 2) \cdot \frac{3}{2u}$

Cancel the matching denominator factors: on the left, $(3u - 2)$ cancels, leaving $2u \cdot 5 = 10u$ . On the right, $2u$ cancels, leaving $(3u - 2) \cdot 3 = 9u - 6$ :

$10u = 9u - 6$

Step 4 — Solve the resulting equation. Subtract $9u$ from both sides:

$u = -6$

Step 5 — Check. The value $u = -6$ does not equal either restricted value ( $0$ or $\frac{2}{3}$ ), so it is not extraneous. Substitute into the original equation:

$\frac{5}{3(-6) - 2} = \frac{5}{-20} = -\frac{1}{4}$ and $\frac{3}{2(-6)} = \frac{3}{-12} = -\frac{1}{4}$

Since $-\frac{1}{4} = -\frac{1}{4}$ is true, $u = -6$ is confirmed as the solution.

This example illustrates a rational equation where both denominators are distinct variable expressions — one a binomial $(3u - 2)$ and one a monomial $2u$ . The LCD is the product of both denominators, and unlike Example 8.60, the resulting equation after clearing fractions is linear rather than quadratic, yielding a single solution.

Updated 2026-04-21

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OpenStax Elementary Algebra

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