11.11 Algebraic Fractions

**Addition.** Simplify $$\frac{3}{x} + \frac{2}{x + 1}$$ (a) Find a common denominator. (b) Combine into a single fraction. (c) State any restrictions on $x$.

**Subtraction.** Simplify $$\frac{4}{x - 1} - \frac{3}{x + 2}$$ (a) Combine as a single fraction. (b) State restrictions on $x$.

**Multiplication & division.** Simplify (a) $\dfrac{x + 1}{x} \cdot \dfrac{2x}{x + 1}$ (b) $\dfrac{x^2}{x + 1} \div \dfrac{x}{x + 1}$ (c) $\dfrac{x^2 - 4}{x - 2} \cdot \dfrac{1}{x + 2}$

**Simplify with factoring.** Simplify fully $$\frac{x^2 - 9}{x^2 - x - 6}$$ State any restrictions on $x$.

**Rational equation.** Solve $$\frac{2}{x - 1} + \frac{3}{x + 2} = 1$$ (a) Multiply through by $(x - 1)(x + 2)$. (b) Solve the resulting quadratic. (c) Check each candidate against the original equation.

**Complex fraction.** Simplify fully $$\frac{\dfrac{1}{x} - \dfrac{1}{x + 1}}{\dfrac{1}{x(x + 1)}}$$

**Partial fractions [EXT].** Find constants $A$ and $B$ such that $$\frac{5x + 1}{(x - 1)(x + 3)} = \frac{A}{x - 1} + \frac{B}{x + 3}$$

**Speed–time.** A car drives the first 60 km of a trip at $v$ km/h, then the next 40 km at $v - 10$ km/h. The total time is 2 hours. (a) Write an equation in $v$. (b) Multiply through to clear fractions, and use the quadratic formula. (c) Give $v$ to 3 s.f.

**Pipe rate.** Pipe A fills a swimming pool in 8 hours; pipe B in 12 hours. (a) What fraction does each pipe fill in 1 hour? (b) Write an equation for $T$, the time for both pipes together. (c) Solve.

**Identity.** Find constants $A$ and $k$ such that $$\frac{3x - 7}{x - 2} = A + \frac{k}{x - 2}$$ for all $x \neq 2$.

**Restrictions.** Simplify fully and state restrictions on $x$. (a) $\dfrac{x^2 - 1}{x^2 + 2x + 1}$ (b) $\dfrac{x^2 - 4x + 4}{x^2 - 4}$ (c) $\dfrac{x^3 - x}{x^2 - 1}$

**Rational equation modelling.** A photo print costs $a$ francs each, and developing a roll of film costs $b$ francs. The average cost per print on a roll with $n$ prints is $\frac{na + b}{n}$. (a) Write the average cost as $a + \frac{b}{n}$. (b) If $a = 0.40$ and $b = 6$, find the number of prints needed for the average cost to be at most CHF 0.70. (c) Comment on what happens to the average cost as $n \to \infty$.