Algebraic fractions

Simplifying algebraic fractions by factorising and cancelling, multiplying and dividing, adding and subtracting with a common denominator, and solving equations with algebraic fractions.

Simplifying

Factorise first, then cancel factors (never single terms): $\dfrac{x^2 - 9}{x + 3} = \dfrac{(x - 3)(x + 3)}{x + 3} = x - 3$.

Multiplying and dividing

Example 1●●

Simplify $\dfrac{x}{3} \times \dfrac{6}{x^2}$ and $\dfrac{x + 1}{4} \div \dfrac{x + 1}{8}$.

$\dfrac{6x}{3x^2} = \dfrac{2}{x}$

$\dfrac{x + 1}{4} \times \dfrac{8}{x + 1} = 2$

Adding and subtracting

Example 2●●●

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

$\dfrac{3(x + 1) - 2(x - 3)}{6}$

$= \dfrac{3x + 3 - 2x + 6}{6} = \dfrac{x + 9}{6}$

Example 3●●●

Solve $\dfrac{x + 1}{2} - \dfrac{x - 3}{3} = 2$.

From Example 2: $\dfrac{x + 9}{6} = 2$

$x + 9 = 12$, so $x = 3$

Flashcards

Cancel only

Factors, not terms.

Dividing fractions

Multiply by the reciprocal.

$dfrac{2}{x} + dfrac{3}{x}$

$\dfrac{5}{x}$

Practice

Simplify $\dfrac{x^2 - 9}{x + 3}$.
Simplify $\dfrac{x}{3} \times \dfrac{6}{x^2}$.
Simplify $\dfrac{x + 1}{4} \div \dfrac{x + 1}{8}$.
Simplify $\dfrac{x + 1}{2} - \dfrac{x - 3}{3}$.
Solve $\dfrac{x + 1}{2} - \dfrac{x - 3}{3} = 2$.
Simplify $\dfrac{x}{3} + \dfrac{x}{4}$.
Simplify $\dfrac{2x + 6}{4}$.
Simplify $\dfrac{2}{x} + \dfrac{3}{x}$.