1Cademy - Subtracting $$\frac{5c+4}{c-2}

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How to Add or Subtract Rational Expressions

Example

Subtracting $\frac{5c+4}{c-2} - 3$

Subtract a whole number from a rational expression by first rewriting the whole number as a fraction with denominator $1$ :

$\frac{5c + 4}{c - 2} - 3$

Step 1 — Write the whole number as a fraction. Express $3$ as $\frac{3}{1}$ so that both terms are rational expressions:

$\frac{5c + 4}{c - 2} - \frac{3}{1}$

Step 2 — Find the LCD and rewrite. The denominators are $(c - 2)$ and $1$ . Since any expression divided by $1$ is itself, the LCD is simply $(c - 2)$ . The first fraction already has the LCD. Multiply the numerator and denominator of the second fraction by $(c - 2)$ :

$\frac{5c + 4}{c - 2} - \frac{3(c - 2)}{1 \cdot (c - 2)}$

Distribute in the second numerator: $3(c - 2) = 3c - 6$ :

$\frac{5c + 4}{c - 2} - \frac{3c - 6}{c - 2}$

Step 3 — Subtract the numerators over the common denominator. Place the second numerator in parentheses and distribute the negative sign:

$\frac{5c + 4 - (3c - 6)}{c - 2} = \frac{5c + 4 - 3c + 6}{c - 2} = \frac{2c + 10}{c - 2}$

Step 4 — Factor to check for common factors. Extract the GCF from the numerator: $2c + 10 = 2(c + 5)$ :

$\frac{2(c + 5)}{c - 2}$

Neither $2$ nor $(c + 5)$ matches the denominator factor $(c - 2)$ , so the expression is already fully simplified.

When one of the expressions in a subtraction problem is not written as a fraction, the first step is to rewrite it as a fraction with denominator $1$ . This converts the problem into a subtraction of two rational expressions, after which the standard three-step procedure — finding the LCD, combining the numerators, and simplifying — applies. Since any expression has $1$ as a factor of its "denominator," the LCD is always just the other fraction's denominator.

Updated 2026-04-21

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OpenStax Elementary Algebra

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