Simplify $\frac{3}{x} + \frac{5}{2x}$.

$\frac{11}{2x}$, with $x \ne 0$.

Simplify $\frac{x^2 - 4}{x + 3} \times \frac{x^2 + 3x}{x - 2}$.

$x(x + 2)$, with $x \ne -3, 2$.

Simplify $\frac{x^2 - 25}{x^2 + 2x} \div \frac{x - 5}{x^2 + 3x + 2}$.

$\frac{(x + 5)(x + 1)}{x}$, with $x \ne 0, -1, -2, 5$.

Simplify $\frac{2}{x - 3} - \frac{1}{x + 2}$.

$\frac{x + 7}{(x - 3)(x + 2)}$, with $x \ne 3, -2$.

Simplify $\frac{5}{x^2 + x - 6} - \frac{4}{x^2 - 4}$ and state the restrictions.

$\frac{1}{(x + 2)(x + 3)}$, with $x \ne 2, -2, -3$.

Grade 11 · MCR3U · Functions

MCR3U: Operations with Rational Expressions

Operations with rational expressions follow the fraction rules you already know. Multiply straight across, divide by multiplying by the reciprocal, and add or subtract over a common denominator. In all four, factoring does the heavy lifting.

How It Works

Multiplying. Factor every numerator and denominator and state the restrictions. Then cancel any factor on top against a matching factor on the bottom, even across the two fractions, and multiply what is left.

Dividing. Flip the second fraction and multiply. The restrictions now come from three places: both original denominators, and the numerator of the fraction you flipped, because it becomes a denominator. Missing that third set is the most common lost mark.

Adding and subtracting need a common denominator. Use the lowest common denominator, the LCD: factor each denominator, then take every factor the greatest number of times it appears in any one of them. For $\frac{1}{x(x + 1)}$ and $\frac{1}{(x + 1)^2}$ the LCD is $x(x + 1)^2$ .

Build each fraction up. Multiply its top and bottom by whatever factors of the LCD it is missing, then combine the numerators over the LCD.

Subtract the whole numerator. Put the second numerator in brackets before the minus sign reaches it: $\frac{x}{x - 2} - \frac{x + 1}{x - 2} = \frac{x - (x + 1)}{x - 2} = \frac{-1}{x - 2}$ . Without the brackets the $1$ keeps the wrong sign.

Simplify at the end. Expand and collect the numerator, then factor it to see whether anything cancels with the LCD. It sometimes does, and a test question is often built so that it will. Keep the restrictions from the original denominators.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Simplify $\frac{3}{x} + \frac{5}{2x}$ .
Show solution
1. The LCD is $2x$ . Build up the first fraction: $\frac{3}{x} = \frac{6}{2x}$ .
2. Add the numerators: $\frac{6 + 5}{2x}$ .
Answer: $\frac{11}{2x}$ , with $x \ne 0$ .
Easy
Simplify $\frac{x^2 - 4}{x + 3} \times \frac{x^2 + 3x}{x - 2}$ .
Show solution
1. Factor: $\frac{(x - 2)(x + 2)}{x + 3} \times \frac{x(x + 3)}{x - 2}$ .
2. Restrictions: $x \ne -3$ and $x \ne 2$ .
3. Cancel $(x - 2)$ and $(x + 3)$ across the fractions.
Answer: $x(x + 2)$ , with $x \ne -3, 2$ .
Medium
Simplify $\frac{x^2 - 25}{x^2 + 2x} \div \frac{x - 5}{x^2 + 3x + 2}$ .
Show solution
1. Factor: $\frac{(x - 5)(x + 5)}{x(x + 2)} \div \frac{x - 5}{(x + 1)(x + 2)}$ .
2. Flip and multiply: $\frac{(x - 5)(x + 5)}{x(x + 2)} \times \frac{(x + 1)(x + 2)}{x - 5}$ .
3. Restrictions: $x \ne 0, -2$ from the first denominator, $x \ne -1, -2$ from the second, and $x \ne 5$ from the numerator that was flipped.
4. Cancel $(x - 5)$ and $(x + 2)$ .
Answer: $\frac{(x + 5)(x + 1)}{x}$ , with $x \ne 0, -1, -2, 5$ .
Medium
Simplify $\frac{2}{x - 3} - \frac{1}{x + 2}$ .
Show solution
1. The LCD is $(x - 3)(x + 2)$ .
2. Build up: $\frac{2(x + 2) - (x - 3)}{(x - 3)(x + 2)}$ .
3. Expand the numerator: $2x + 4 - x + 3 = x + 7$ .
Answer: $\frac{x + 7}{(x - 3)(x + 2)}$ , with $x \ne 3, -2$ .
Test level
Simplify $\frac{5}{x^2 + x - 6} - \frac{4}{x^2 - 4}$ and state the restrictions.
Show solution
1. Factor the denominators: $x^2 + x - 6 = (x + 3)(x - 2)$ and $x^2 - 4 = (x - 2)(x + 2)$ .
2. Restrictions: $x \ne -3, 2, -2$ . The LCD is $(x - 2)(x + 2)(x + 3)$ .
3. Build up: $\frac{5(x + 2) - 4(x + 3)}{(x - 2)(x + 2)(x + 3)}$ .
4. The numerator is $5x + 10 - 4x - 12 = x - 2$ , which cancels with the factor $(x - 2)$ in the LCD.
Answer: $\frac{1}{(x + 2)(x + 3)}$ , with $x \ne 2, -2, -3$ .

Learn It on Martian Lab

Martian Lab teaches this in the Adding & Subtracting Rational Expressions lesson of its MCR3U Rational Expressions unit. Watch it, then take the Rational Expressions practice tests to see if it stuck.

Watch the Adding & Subtracting Rational Expressions lesson free

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Free Rational Expressions practice test, with solutions

More Rational Expressions topics: Simplifying Rational Expressions

More MCR3U topics: Inverse Functions · Transformations of Functions · Sine Law and the Ambiguous Case · Domain, Range and Function Notation · Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems · Simplifying Radicals · Exponent Laws and Rational Exponents · Graphing Exponential Functions · Exponential Growth and Decay · Special Angles and the CAST Rule · Cosine Law and 3D Problems · Transformations of Sine and Cosine · Sinusoidal Word Problems · Arithmetic Sequences and Series · Geometric Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

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