Corrigé
11.11 Algebraic Fractions
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Simplify $\dfrac{ 6x}{ 9x}$ (assume $x \neq 0$). | $\dfrac{2}{3}$ |
| 2 | Simplify $\dfrac{ 6x + 9}{ 3}$. | $2x + 3$ |
| 3 | Simplify $\dfrac{ 3}{x} + \dfrac{ 5}{x}$. | $\dfrac{8}{x}$ |
| 4 | Simplify $\dfrac{ 4}{x} \cdot \dfrac{x}{ 3}$. | $\dfrac{4}{3}$ |
| 5 | Solve $\dfrac{x}{ 4} = 5$. | $x = 20$ |
| 6 | Simplify $\dfrac{ 6}{x} \div \dfrac{ 3}{x}$. | 2 |
| 7 | Simplify $\dfrac{x^2}{x}$ (assume $x \neq 0$). | $x$ |
| 8 | State the value of $x$ for which $\dfrac{1}{x - 5}$ is undefined. | $x = 5$ |
| 9 | Evaluate $\dfrac{x + 1}{x - 1}$ at $x = 3$. | 2 |
| 10 | State the values of $x$ for which $\dfrac{x + 2}{x(x - 3)}$ is undefined. | $x = 0$ and $x = 3$ |
| 11 | Simplify $\dfrac{2}{x} + \dfrac{3}{ 5}$ as a single fraction. | $\dfrac{10 + 3x}{5x}$ |
| 12 | Simplify $\dfrac{ 3}{x} - \dfrac{ 2}{x + 1}$ as a single fraction. | $\dfrac{x + 3}{x(x + 1)}$ |
| 13 | Simplify $\dfrac{ 4x}{ 9} \cdot \dfrac{ 6}{ 2x^2}$. | $\dfrac{4}{3x}$ |
| 14 | Solve $\dfrac{x}{2} + \dfrac{x}{3} = 10$. | $x = 12$ |
| 15 | Solve $\dfrac{x}{x + 1} = \dfrac{ 2}{ 3}$. | $x = 2$ |
| 16 | Simplify $\dfrac{x^2 - 3^2}{x^2 - 3x}$. | $\dfrac{x + 3}{x}$ |
| 17 | Simplify $\dfrac{1}{x} + \dfrac{2}{x + 1}$ as a single fraction. | $\dfrac{3x + 1}{x(x + 1)}$ |
| 18 | Simplify and state restrictions: $\dfrac{(x - 2)(x + 3)}{(x + 3)(x - 5)}$. | $\dfrac{x - 2}{x - 5}$; $x \neq -3$ (and $x \neq 5$ for the simplified expression) |
| 19 | Solve $\dfrac{2}{x - 1} = \dfrac{3}{x + 2}$. | $x = 7$ |
| 20 | Simplify $\dfrac{3}{x - 2} - \dfrac{2}{x + 1}$ as a single fraction. | $\dfrac{x + 7}{(x - 2)(x + 1)}$ |
| 21 | Simplify $\dfrac{1}{x} - \dfrac{1}{x + 1} + \dfrac{1}{x(x + 1)}$. | $\dfrac{2}{x(x + 1)}$ |
| 22 | Simplify $\dfrac{x^2 + 5x + 6}{x^2 + x - 6}$. | $\dfrac{x + 2}{x - 2}$ |
| 23 | Simplify $\dfrac{\dfrac{1}{x} + 1}{1 - \dfrac{1}{x}}$. | $\dfrac{x + 1}{x - 1}$ |
| 24 | Solve $\dfrac{2}{x} + \dfrac{3}{x + 1} = 2$. | $x = -\dfrac{1}{2}$ or $x = 2$ |
| 25 | Find $k$ such that $\dfrac{2x + k}{x - 3} = 2 + \dfrac{11}{x - 3}$. | $k = 5$ |
| 26 | Simplify $\dfrac{(x + 2)^2 - 4}{x}$. | $x + 4$ |
| 27 | Solve $\dfrac{x - 2}{3} - \dfrac{x + 1}{2} = 1$. | $x = -13$ |
| 28 | Simplify $\dfrac{\sqrt{x}(\sqrt{x} + 1)}{x}$. | $1 + \dfrac{1}{\sqrt{x}}$ |
| 29 | Solve $\dfrac{x + 3}{x - 2} = 2$. | $x = 7$ |
| 30 | Simplify $\dfrac{x^2 - 9}{x^2 + 6x + 9}$ and state restrictions on $x$. | $\dfrac{x - 3}{x + 3}$; $x \neq -3$ |
| 31 | Solve $\dfrac{2}{x - 1} + \dfrac{3}{x + 2} = 1$ and verify each solution against the original equation. | $x = 2 \pm \sqrt{7}$ |
| 32 | Find constants $A$ and $B$ such that $\dfrac{3x + 5}{(x + 1)(x + 2)} = \dfrac{A}{x + 1} + \dfrac{B}{x + 2}$. | $A = 2$, $B = 1$ |
| 33 | Simplify $\dfrac{\dfrac{x}{x - 1} - 1}{\dfrac{1}{x - 1} + 1}$. | $\dfrac{1}{x}$ |
| 34 | Solve $\dfrac{1}{x - 1} - \dfrac{2}{x^2 - 1} = \dfrac{1}{x + 1}$ and state restrictions. | Identity (true for all $x \neq \pm 1$) |
| 35 | Solve $\dfrac{x^2 - 4}{x - 2} + \dfrac{x^2 - 9}{x - 3} = 14$. | $x = 5$ |
| 36 | Pipe A fills a tank in $x$ hours; pipe B fills it in $x + 3$ hours. Together they fill it in 2 hours. Find $x$. | $x = 3$ |
| 37 | Solve $\;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \; \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\;$ for $x, y$. | $x = 2$, $y = 3$ |
| 38 | Simplify $\dfrac{2}{x - 1} - \dfrac{3}{x + 1} + \dfrac{4x}{x^2 - 1}$. | $\dfrac{3x + 5}{x^2 - 1}$ |
| 39 | Find $A$, $B$, $C$ such that $\dfrac{3x^2 + 1}{x(x - 1)(x + 1)} = \dfrac{A}{x} + \dfrac{B}{x - 1} + \dfrac{C}{x + 1}$. | $A = -1$, $B = 2$, $C = 2$ |
| 40 | A car drives the first 60 km at $v$ km/h then the next 40 km at $v - 10$ km/h. The total time is 2 hours. Write the equation and find $v$ to 3 s.f. | $v \approx 54.5$ km/h |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Simplify $\dfrac{ 8x}{ 12x}$ (assume $x \neq 0$). | $\dfrac{2}{3}$ |
| 2 | Simplify $\dfrac{ 10x + 15}{ 5}$. | $2x + 3$ |
| 3 | Simplify $\dfrac{ 2}{x} + \dfrac{ 7}{x}$. | $\dfrac{9}{x}$ |
| 4 | Simplify $\dfrac{ 5}{x} \cdot \dfrac{x}{ 2}$. | $\dfrac{5}{2}$ |
| 5 | Solve $\dfrac{x}{ 3} = 7$. | $x = 21$ |
| 6 | Simplify $\dfrac{ 8}{x} \div \dfrac{ 4}{x}$. | 2 |
| 7 | Simplify $\dfrac{x^2}{x}$ (assume $x \neq 0$). | $x$ |
| 8 | State the value of $x$ for which $\dfrac{1}{x - 5}$ is undefined. | $x = \dfrac{3}{2}$ |
| 9 | Evaluate $\dfrac{x + 1}{x - 1}$ at $x = 3$. | $\dfrac{1}{3}$ |
| 10 | State the values of $x$ for which $\dfrac{x + 2}{x(x - 3)}$ is undefined. | $x = -4$ and $x = 2$ |
| 11 | Simplify $\dfrac{2}{x} + \dfrac{3}{ 4}$ as a single fraction. | $\dfrac{8 + 3x}{4x}$ |
| 12 | Simplify $\dfrac{ 4}{x} - \dfrac{ 3}{x + 1}$ as a single fraction. | $\dfrac{x + 4}{x(x + 1)}$ |
| 13 | Simplify $\dfrac{ 3x}{ 8} \cdot \dfrac{ 4}{ 6x^2}$. | $\dfrac{1}{4x}$ |
| 14 | Solve $\dfrac{x}{2} + \dfrac{x}{3} = 15$. | $x = 18$ |
| 15 | Solve $\dfrac{x}{x + 1} = \dfrac{ 3}{ 5}$. | $x = \dfrac{3}{2}$ |
| 16 | Simplify $\dfrac{x^2 - 4^2}{x^2 - 4x}$. | $\dfrac{x + 4}{x}$ |
| 17 | Simplify $\dfrac{1}{x} + \dfrac{2}{x + 1}$ as a single fraction. | $\dfrac{2x}{x^2 - 1}$ |
| 18 | Simplify and state restrictions: $\dfrac{(x - 2)(x + 3)}{(x + 3)(x - 5)}$. | $\dfrac{x + 1}{x + 6}$; $x \neq 4$ and $x \neq -6$ |
| 19 | Solve $\dfrac{2}{x - 1} = \dfrac{3}{x + 2}$. | $x = -7$ |
| 20 | Simplify $\dfrac{3}{x - 2} - \dfrac{2}{x + 1}$ as a single fraction. | $\dfrac{x - 13}{(x + 3)(x - 1)}$ |
| 21 | Simplify $\dfrac{1}{x} - \dfrac{1}{x + 1} + \dfrac{1}{x(x + 1)}$. | $\dfrac{2}{x + 1}$ |
| 22 | Simplify $\dfrac{x^2 + 5x + 6}{x^2 + x - 6}$. | $\dfrac{x - 2}{x + 2}$ |
| 23 | Simplify $\dfrac{\dfrac{1}{x} + 1}{1 - \dfrac{1}{x}}$. | $\dfrac{x + 2}{x - 2}$ |
| 24 | Solve $\dfrac{2}{x} + \dfrac{3}{x + 1} = 2$. | $x = 2 \pm \sqrt{6}$ |
| 25 | Find $k$ such that $\dfrac{2x + k}{x - 3} = 2 + \dfrac{11}{x - 3}$. | $k = 2$ |
| 26 | Simplify $\dfrac{(x + 2)^2 - 4}{x}$. | $x - 6$ |
| 27 | Solve $\dfrac{x - 2}{3} - \dfrac{x + 1}{2} = 1$. | $x = 4$ |
| 28 | Simplify $\dfrac{\sqrt{x}(\sqrt{x} + 1)}{x}$. | $\sqrt{x} - 2$ |
| 29 | Solve $\dfrac{x + 3}{x - 2} = 2$. | $x = -\dfrac{13}{2}$ |
| 30 | Simplify $\dfrac{x^2 - 9}{x^2 + 6x + 9}$ and state restrictions on $x$. | $\dfrac{x + 4}{x + 2}$; $x \neq \pm 2$ |
| 31 | Solve $\dfrac{2}{x - 1} + \dfrac{3}{x + 2} = 1$ and verify each solution against the original equation. | $x = 1 \pm \sqrt{4}$ — i.e. $x = -1$ or $x = 3$; but $x = 3$ makes a denominator zero, so reject. $x = -1$. |
| 32 | Find constants $A$ and $B$ such that $\dfrac{3x + 5}{(x + 1)(x + 2)} = \dfrac{A}{x + 1} + \dfrac{B}{x + 2}$. | $A = \dfrac{3}{4}$, $B = \dfrac{5}{4}$ |
| 33 | Simplify $\dfrac{\dfrac{x}{x - 1} - 1}{\dfrac{1}{x - 1} + 1}$. | $\dfrac{2 - x}{2 + x}$ |
| 34 | Solve $\dfrac{1}{x - 1} - \dfrac{2}{x^2 - 1} = \dfrac{1}{x + 1}$ and state restrictions. | Solve: $(x - 2) + 1 = (x + 2) \Rightarrow -1 = 0$, false. No solution. |
| 35 | Solve $\dfrac{x^2 - 4}{x - 2} + \dfrac{x^2 - 9}{x - 3} = 14$. | $x = -1$ |
| 36 | Pipe A fills a tank in $x$ hours; pipe B fills it in $x + 3$ hours. Together they fill it in 2 hours. Find $x$. | $x = 5$ |
| 37 | Solve $\;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \; \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\;$ for $x, y$. | $x = 3$, $y = 4$ |
| 38 | Simplify $\dfrac{2}{x - 1} - \dfrac{3}{x + 1} + \dfrac{4x}{x^2 - 1}$. | $\dfrac{2x + 12}{x^2 - 4}$ |
| 39 | Find $A$, $B$, $C$ such that $\dfrac{3x^2 + 1}{x(x - 1)(x + 1)} = \dfrac{A}{x} + \dfrac{B}{x - 1} + \dfrac{C}{x + 1}$. | $A = -1$, $B = 1$, $C = 1$ |
| 40 | A car drives the first 60 km at $v$ km/h then the next 40 km at $v - 10$ km/h. The total time is 2 hours. Write the equation and find $v$ to 3 s.f. | $v \approx 50.0$ km/h |
Problèmes — Solutions détaillées
**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 $x$.
(a) $x(x + 1)$. (b) $\dfrac{5x + 3}{x(x + 1)}$. (c) $x \neq 0$ and $x \neq -1$.
**Subtraction.** Simplify $$\frac{4}{x - 1} - \frac{3}{x + 2}$$ (a) Combine as a single fraction. (b) State restrictions on $x$.
(a) $\dfrac{x + 11}{(x - 1)(x + 2)}$. (b) $x \neq 1$, $x \neq -2$.
**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) $\dfrac{x^2 - 4}{x - 2} \cdot \dfrac{1}{x + 2}$
(a) 2. (b) $x$. (c) 1 (with restrictions $x \neq \pm 2$).
**Simplify with factoring.** Simplify fully $$\frac{x^2 - 9}{x^2 - x - 6}$$ State any restrictions on $x$.
$\dfrac{x + 3}{x + 2}$, with $x \neq 3$ and $x \neq -2$.
**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.
$x = 2 + \sqrt{7}$ or $x = 2 - \sqrt{7}$.
**Complex fraction.** Simplify fully $$\frac{\dfrac{1}{x} - \dfrac{1}{x + 1}}{\dfrac{1}{x(x + 1)}}$$
$1$.
**Partial fractions [EXT].** Find constants $A$ and $B$ such that $$\frac{5x + 1}{(x - 1)(x + 3)} = \frac{A}{x - 1} + \frac{B}{x + 3}$$
$A = \dfrac{3}{2}$, $B = \dfrac{7}{2}$.
**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.
(a) $\frac{60}{v} + \frac{40}{v - 10} = 2$. (b) $v^2 - 60v + 300 = 0$. (c) $v \approx 54.5$ km/h.
**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.
(a) A: $\frac{1}{8}$; B: $\frac{1}{12}$. (b) $\frac{1}{8} + \frac{1}{12} = \frac{1}{T}$. (c) $T = 4.8$ h.
**Identity.** Find constants $A$ and $k$ such that $$\frac{3x - 7}{x - 2} = A + \frac{k}{x - 2}$$ for all $x \neq 2$.
$A = 3$, $k = -1$.
**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) $\dfrac{x^3 - x}{x^2 - 1}$
(a) $\dfrac{x - 1}{x + 1}$; $x \neq -1$. (b) $\dfrac{x - 2}{x + 2}$; $x \neq \pm 2$. (c) $x$; $x \neq \pm 1$.
**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 $n \to \infty$.
(a) See working. (b) $n \geq 20$. (c) Approaches $a = 0.40$ from above.