How to Add and Subtract Rational Expressions? (+FREE Worksheet!)

Adding and subtracting rational expressions follows the same logic as adding and subtracting ordinary fractions: you need a common denominator before you can combine the numerators. The key extra step is finding the least common denominator (LCD) of the algebraic expressions, and once you do, combining the numerators and simplifying is straightforward. This skill is essential in Algebra 1 and used extensively in higher-level algebra courses.

What Are Rational Expressions?

A rational expression is a fraction whose numerator and denominator are polynomials. Examples: \(\color{blue}{\frac{3}{(x + 1)}}\), \(\color{blue}{\frac{(x – 2)}{(x^{2} – 4)}}\), and \(\color{blue}{\frac{5}{x}}\). Adding and subtracting rational expressions is exactly like adding fractions — you need a common denominator first.

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How to Add and Subtract Rational Expressions

Case 1 — Same denominators

If the denominators are identical, simply add or subtract the numerators over the common denominator, then simplify.

Example: \(\color{blue}{\frac{2}{(x + 1)} + \frac{3}{(x + 1)} = \frac{(2 + 3)}{(x + 1)}}\) = \(\color{blue}{\frac{5}{(x + 1)}}\)

Case 2 — Different denominators

Find the LCD, build equivalent fractions with that LCD, then add or subtract the numerators.

Example: \(\color{blue}{\frac{1}{(x – 2)} + \frac{2}{(x + 3)}}\).
\(\color{blue}{\text{ LCD } = (x – 2)(x + 3)}\).
Rewrite: \(\color{blue}{\frac{(1(x + 3) + 2(x – 2))}{((x – 2)(x + 3))}}\)
= \(\color{blue}{\frac{(x + 3 + 2x – 4)}{((x – 2)(x + 3))}}\)
= \(\color{blue}{\frac{(3x – 1)}{((x – 2)(x + 3))}}\)

How to find the LCD

Factor each denominator. The LCD is the product of each distinct factor raised to its highest power. Always factor first.

Step-by-Step Summary

1. Factor each denominator completely.
2. Identify the LCD (product of all distinct factors at their highest powers).
3. Rewrite each fraction as an equivalent fraction with the LCD.
4. Add or subtract the numerators; keep the LCD as the denominator.
5. Simplify the resulting numerator and reduce the fraction if possible.

Watch: Adding and Subtracting Rational Expressions with Unlike Denominators (Video Lesson)

The Organic Chemistry Tutor demonstrates how to add and subtract rational expressions with unlike denominators step by step:

Adding and Subtracting Rational Expressions – Worked Examples

Example 1: Simplify \(\color{blue}{\frac{2}{(x + 1)} + \frac{3}{(x + 1)}}\).

Same denominators: \(\color{blue}{\frac{(2 + 3)}{(x + 1)}}\) = \(\color{blue}{\frac{5}{(x + 1)}}\)

Example 2: Simplify \(\color{blue}{\frac{1}{(x + 2)} + \frac{1}{(x – 2)}}\).

\(\color{blue}{\text{ LCD } = (x + 2)(x – 2) = x}\)\(\color{blue}{^{2} – 4}\).
= \(\color{blue}{\frac{((x – 2) + (x + 2))}{(x^{2} – 4)}}\)
= \(\color{blue}{\frac{(2x)}{(x^{2} – 4)}}\) = \(\color{blue}{\frac{2x}{(x^{2} – 4)}}\)

Example 3: Simplify \(\color{blue}{\frac{3}{(x + 1)} – \frac{2}{(x – 1)}}\).

\(\color{blue}{\text{ LCD } = (x + 1)(x – 1)}\).
= \(\color{blue}{\frac{(3(x – 1) – 2(x + 1))}{((x + 1)(x – 1))}}\)
= \(\color{blue}{\frac{(3x – 3 – 2x – 2)}{((x + 1)(x – 1))}}\)
= \(\color{blue}{\frac{(x – 5)}{(x^{2} – 1)}}\)

Example 4: Simplify \(\color{blue}{\frac{1}{(x – 2)} + \frac{2}{(x + 3)}}\).

\(\color{blue}{\text{ LCD } = (x – 2)(x + 3)}\).
= \(\color{blue}{\frac{((x + 3) + 2(x – 2))}{((x – 2)(x + 3))}}\)
= \(\color{blue}{\frac{(x + 3 + 2x – 4)}{((x – 2)(x + 3))}}\)
= \(\color{blue}{\frac{(3x – 1)}{((x – 2)(x + 3))}}\)

More Practice: Adding and Subtracting Rational Expressions Video (Khan Academy)

Khan Academy explains the process of adding and subtracting rational expressions with additional examples and interactive practice:

Exercises for Adding and Subtracting Rational Expressions

1. \(\color{blue}{\frac{4}{(x – 3)} + \frac{2}{(x – 3)}}\)
2. \(\color{blue}{\frac{1}{(x + 5)} – \frac{1}{(x – 5)}}\)
3. \(\color{blue}{\frac{3}{(x – 1)} – \frac{1}{(x + 2)}}\)
4. \(\color{blue}{\frac{2}{(x + 4)} + \frac{3}{(x – 4)}}\)
5. \(\color{blue}{\frac{x}{(x^{2} – 4)} + \frac{(x – 1)}{(x + 2)}}\)

Answers

1. Same denominator: \(\color{blue}{\frac{6}{(x – 3)}}\)
2. \(\color{blue}{\text{ LCD } = (x + 5)(x – 5)}\): \(\color{blue}{\frac{((x – 5) – (x + 5))}{((x + 5)(x – 5))}}\) = −\(\color{blue}{\frac{10}{(x^{2} – 25)}}\) = −\(\color{blue}{\frac{10}{(x^{2} – 25)}}\)
3. \(\color{blue}{\text{ LCD } = (x – 1)(x + 2)}\): \(\color{blue}{\frac{(3(x + 2) – (x – 1))}{((x – 1)(x + 2))} = \frac{(2x + 7)}{(x^{2} + x – 2)}}\) = \(\color{blue}{\frac{(2x + 7)}{((x – 1)(x + 2))}}\)
4. \(\color{blue}{\text{ LCD } = (x + 4)(x – 4)}\): \(\color{blue}{\frac{(2(x – 4) + 3(x + 4))}{(x^{2} – 16)} = \frac{(5x + 4)}{(x^{2} – 16)}}\) = \(\color{blue}{\frac{(5x + 4)}{(x^{2} – 16)}}\)
5. x² − \(\color{blue}{4 = (x + 2)(x – 2)}\); \(\color{blue}{\text{ LCD } = (x + 2)(x – 2)}\):
   \(\color{blue}{\frac{(x + (x – 1)(x – 2))}{((x + 2)(x – 2))} = \frac{(x + x^{2} – 3x + 2)}{(x^{2} – 4)}}\) = \(\color{blue}{\frac{(x^{2} – 2x + 2)}{(x^{2} – 4)}}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Adding and Subtracting Rational Expressions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work. They’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Adding and Subtracting Polynomials Worksheet
• Download Simplifying Algebraic Expressions Worksheet

Frequently Asked Questions

What steps assist with adding and subtracting rational expressions?

Five steps keep these problems organized. Factor every denominator. Find the least common denominator (LCD). Rewrite each fraction with the LCD. Combine the numerators, putting the subtracted numerator in parentheses. Finally, simplify and list the values that make any original denominator zero. Even a simple case follows this path: x/4 + x/6 = 3x/12 + 2x/12 = 5x/12.

How do you add and subtract rational expressions with like denominators?

When the denominators already match, combine the numerators and keep the denominator. For 3/(x + 1) + 5/(x + 1), the answer is 8/(x + 1). With subtraction, the minus sign applies to the whole second numerator: (2x + 5)/(x − 2) − (x + 3)/(x − 2) = (2x + 5 − x − 3)/(x − 2) = (x + 2)/(x − 2), with x ≠ 2.

How do you add and subtract rational expressions with unlike denominators?

Build a common denominator from the factors of each denominator, then rewrite and combine. For 2/x + 3/(x + 1), the LCD is x(x + 1). Multiply 2/x by (x + 1)/(x + 1) and 3/(x + 1) by x/x. The numerators become 2(x + 1) + 3x = 5x + 2, so the sum is (5x + 2)/(x(x + 1)), with x ≠ 0 and x ≠ −1.

Is adding and subtracting rational expressions hard?

It feels hard at first, but it is the same skill as adding fractions like 1/3 + 1/4, with factoring added. Most errors come from three habits: forgetting to distribute a minus sign, canceling terms across a plus sign, and skipping the restrictions. If you can factor trinomials and find a common denominator with numbers, a little practice makes these problems routine.

How do you simplify when adding and subtracting rational expressions?

After combining, factor the new numerator and cancel any factor it shares with the denominator. For 1/(x − 3) − 6/(x² − 9), factor x² − 9 as (x − 3)(x + 3) and use that as the LCD. The numerator becomes (x + 3) − 6 = x − 3, so you get (x − 3)/((x − 3)(x + 3)) = 1/(x + 3), with x ≠ 3 and x ≠ −3.

Related Topics

• Multiplying Rational Expressions
• Dividing Rational Expressions
• Simplifying Complex Fractions
• Graphing Rational Functions
• Finding Domain and Range of a Function