Subtracting Rational Expressions (Like Denominators)

Learn how to subtract rational expressions that share the same denominator by combining numerators.

Advanced25 minLesson
Printable worksheet

Definition

To subtract rational expressions with like denominators, subtract the numerators and keep the common denominator:
AC−BC=A−BC\frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}
where C≠0C \neq 0.
Critical Step: When subtracting, you must distribute the negative sign to every term in the second numerator:
3x+5x−2−x−1x−2=(3x+5)−(x−1)x−2=3x+5−x+1x−2=2x+6x−2\frac{3x + 5}{x - 2} - \frac{x - 1}{x - 2} = \frac{(3x + 5) - (x - 1)}{x - 2} = \frac{3x + 5 - x + 1}{x - 2} = \frac{2x + 6}{x - 2}
Always simplify the final result if possible.

Try it now

What is the first step when subtracting 5xx+2−3xx+2\frac{5x}{x+2} - \frac{3x}{x+2}?

Worked Examples

Simplify: 7xx+3−2xx+3\frac{7x}{x + 3} - \frac{2x}{x + 3}
1

Verify denominators are the same

Both have denominator (x+3)(x + 3) → Like denominators confirmed
2

Subtract the numerators

7x−2x=5x7x - 2x = 5x → 5xx+3\frac{5x}{x + 3}
3

Check if result can be simplified

5x5x and (x+3)(x + 3) share no common factors → Cannot simplify further

Common Mistakes

Forgetting to distribute the negative sign to all terms

Why it's wrong: When you subtract (x−3)(x - 3), the negative applies to both terms: −x-x AND +3+3.

Correct: Always use parentheses: (5x+2)−(x−3)=5x+2−x+3=4x+5(5x + 2) - (x - 3) = 5x + 2 - x + 3 = 4x + 5

Subtracting the denominators too

Why it's wrong: With like denominators, only the numerators change. The denominator stays the same.

Correct: AC−BC=A−BC\frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}, NOT A−BC−C\frac{A - B}{C - C}

Not simplifying the final answer

Why it's wrong: After subtracting, always factor and look for common factors to cancel.

Correct: If you get 2x+4x+2\frac{2x + 4}{x + 2}, factor to 2(x+2)x+2=2\frac{2(x + 2)}{x + 2} = 2

Dropping parentheses before distributing

Why it's wrong: Writing A−BC\frac{A - B}{C} without parentheses around BB can lead to sign errors.

Correct: Always write (A)−(B)C\frac{(A) - (B)}{C} first, then distribute

Watch a video explanation

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the first step when subtracting 5xx+2−3xx+2\frac{5x}{x+2} - \frac{3x}{x+2}?

Why It Matters

Subtracting rational expressions is essential in algebra and beyond:
  • Solving Equations: Many algebraic equations require combining rational expressions
  • Physics: Calculating differences in rates, velocities, or electrical resistances
  • Economics: Finding the difference between cost and revenue functions
  • Engineering: Computing differences in signal frequencies or transfer functions
Mastering this skill builds the foundation for working with complex algebraic expressions.

Real World Applications

Profit Margin Analysis

Businesses calculate net profit margins by subtracting cost ratios from revenue ratios.

Example:

If revenue per unit is 5x+100x\frac{5x + 100}{x} and cost per unit is 3x+60x\frac{3x + 60}{x}, the profit per unit is 5x+100−3x−60x=2x+40x\frac{5x + 100 - 3x - 60}{x} = \frac{2x + 40}{x}.

1Try It Yourself

A company's revenue function is 8x+200x+5\frac{8x + 200}{x + 5} and expenses are 3x+50x+5\frac{3x + 50}{x + 5} per product.

What is the simplified profit expression?

Step 1: Write the mathematical expression

Subtract the expense from revenue:

Physics: Relative Motion

When calculating relative velocities or differences in rates, we often subtract rational expressions.

Example:

If object A moves at dt−2\frac{d}{t-2} and object B at d−10t−2\frac{d-10}{t-2}, their relative position involves d−(d−10)t−2=10t−2\frac{d - (d-10)}{t-2} = \frac{10}{t-2}.

2Try It Yourself

Two signals travel at rates 4t+12t+3\frac{4t + 12}{t + 3} and 2t+6t+3\frac{2t + 6}{t + 3} meters per second.

What is the difference in their rates?

Step 1: Write the mathematical expression

Subtract the second rate from the first:

Key Takeaways

  • 1Subtracting rational expressions with like denominators: AC−BC=A−BC\frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}
  • 2Always use parentheses around the second numerator before distributing the negative sign
  • 3The negative sign must be distributed to EVERY term in the subtracted expression
  • 4After subtracting, combine like terms and simplify by factoring
  • 5Check if the final result can be reduced by canceling common factors

Frequently Asked Questions

Parentheses ensure you distribute the negative sign to all terms. Without them, you might only negate the first term and get the wrong answer.
Parentheses ensure you distribute the negative sign to all terms. Without them, you might only negate the first term and get the wrong answer.
That's fine! As long as there are no common factors between the numerator and denominator, your factored form is the final answer.
Yes! If the numerators are identical, subtracting gives 0C=0\frac{0}{C} = 0 (where C≠0C \neq 0).

Glossary

Rational expression
A fraction where the numerator and/or denominator contain variables (polynomials)
Like denominators
Fractions that have exactly the same expression in their denominators
Distribute
Apply an operation (like negation) to every term inside parentheses
Simplify
Reduce a rational expression by canceling common factors from numerator and denominator

Formula Card

Subtraction with Like Denominators

AC−BC=A−BC\frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}

Subtract numerators, keep the common denominator

Distribute Negative Sign

(A)−(B−C)=A−B+C(A) - (B - C) = A - B + C

The negative changes all signs in the second expression

Today's challengeOne problem, three tries

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