Algebraic fractions and rational equations

(a) $\dfrac{3x}{x - 2}$

• M1 Factorising the numerator to $3x(x + 2)$ and the denominator to $(x + 2)(x - 2)$.
• A1 $\frac{3x}{x - 2}$.

(b) $\dfrac{3}{2x}$

• M1 Multiplying by the reciprocal $\frac{15}{4x^2}$.
• A1 The correct answer, $\frac{3}{2x}$.

(a) $((x - 3)(x - 1))/((x - 2)(x + 3))$

• M1 Establishing $(x-3)(x+3)/((x+3)(x-2))$ or an equivalent valid method.
• M1 Establishing $\frac{x-1}{x+3}$ or an equivalent valid method.
• A1 Correct answer: $((x - 3)(x - 1))/((x - 2)(x + 3))$

(a) $\frac{x(x + 3) - 18}{(x - 3)(x + 3)} = \frac{(x + 6)(x - 3)}{(x - 3)(x + 3)} = \frac{x + 6}{x + 3}$

• P1 Using the common denominator $(x - 3)(x + 3)$.
• P1 Reaching the numerator $x^2 + 3x - 18$.
• M1 Factorising the numerator to $(x + 6)(x - 3)$.
• A1 Cancelling $(x - 3)$ to reach $\frac{x + 6}{x + 3}$.

(b) $x = 0$

• M1 Forming $x + 6 = 2(x + 3)$.
• A1 The correct answer, $x = 0$.