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Fluidité · Pack B

11.11 Algebraic Fractions

Répondez à chaque question. Montrez les calculs si nécessaire.

Bronze

  1. Simplify $\dfrac{ 8x}{ 12x}$ (assume $x \neq 0$).

  2. Simplify $\dfrac{ 10x + 15}{ 5}$.

  3. Simplify $\dfrac{ 2}{x} + \dfrac{ 7}{x}$.

  4. Simplify $\dfrac{ 5}{x} \cdot \dfrac{x}{ 2}$.

  5. Solve $\dfrac{x}{ 3} = 7$.

  6. Simplify $\dfrac{ 8}{x} \div \dfrac{ 4}{x}$.

  7. Simplify $\dfrac{x^2}{x}$ (assume $x \neq 0$).

  8. State the value of $x$ for which $\dfrac{1}{x - 5}$ is undefined.

  9. Evaluate $\dfrac{x + 1}{x - 1}$ at $x = 3$.

  10. State the values of $x$ for which $\dfrac{x + 2}{x(x - 3)}$ is undefined.

Silver

  1. Simplify $\dfrac{2}{x} + \dfrac{3}{ 4}$ as a single fraction.

  2. Simplify $\dfrac{ 4}{x} - \dfrac{ 3}{x + 1}$ as a single fraction.

  3. Simplify $\dfrac{ 3x}{ 8} \cdot \dfrac{ 4}{ 6x^2}$.

  4. Solve $\dfrac{x}{2} + \dfrac{x}{3} = 15$.

  5. Solve $\dfrac{x}{x + 1} = \dfrac{ 3}{ 5}$.

  6. Simplify $\dfrac{x^2 - 4^2}{x^2 - 4x}$.

  7. Simplify $\dfrac{1}{x} + \dfrac{2}{x + 1}$ as a single fraction.

  8. Simplify and state restrictions: $\dfrac{(x - 2)(x + 3)}{(x + 3)(x - 5)}$.

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

  10. Simplify $\dfrac{3}{x - 2} - \dfrac{2}{x + 1}$ as a single fraction.

Gold

  1. Simplify $\dfrac{1}{x} - \dfrac{1}{x + 1} + \dfrac{1}{x(x + 1)}$.

  2. Simplify $\dfrac{x^2 + 5x + 6}{x^2 + x - 6}$.

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

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

  5. Find $k$ such that $\dfrac{2x + k}{x - 3} = 2 + \dfrac{11}{x - 3}$.

  6. Simplify $\dfrac{(x + 2)^2 - 4}{x}$.

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

  8. Simplify $\dfrac{\sqrt{x}(\sqrt{x} + 1)}{x}$.

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

  10. Simplify $\dfrac{x^2 - 9}{x^2 + 6x + 9}$ and state restrictions on $x$.

Platinum

  1. Solve $\dfrac{2}{x - 1} + \dfrac{3}{x + 2} = 1$ and verify each solution against the original equation.

  2. Find constants $A$ and $B$ such that $\dfrac{3x + 5}{(x + 1)(x + 2)} = \dfrac{A}{x + 1} + \dfrac{B}{x + 2}$.

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

  4. Solve $\dfrac{1}{x - 1} - \dfrac{2}{x^2 - 1} = \dfrac{1}{x + 1}$ and state restrictions.

  5. Solve $\dfrac{x^2 - 4}{x - 2} + \dfrac{x^2 - 9}{x - 3} = 14$.

  6. Pipe A fills a tank in $x$ hours; pipe B fills it in $x + 3$ hours. Together they fill it in 2 hours. Find $x$.

  7. Solve $\;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \; \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\;$ for $x, y$.

  8. Simplify $\dfrac{2}{x - 1} - \dfrac{3}{x + 1} + \dfrac{4x}{x^2 - 1}$.

  9. Find $A$, $B$, $C$ such that $\dfrac{3x^2 + 1}{x(x - 1)(x + 1)} = \dfrac{A}{x} + \dfrac{B}{x - 1} + \dfrac{C}{x + 1}$.

  10. 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.