Algebraic fractions questions and answers

Worked solution

1. $x^2 + 5x + 6 = (x + 2)(x + 3)$
2. $\dfrac{(x + 2)(x + 3)}{x + 2} = x + 3$

Worked solution

1. $\dfrac{3(x + 1)}{12} + \dfrac{2(x - 3)}{12}$
2. $= \dfrac{3x + 3 + 2x - 6}{12} = \dfrac{5x - 3}{12}$

Worked solution

1. $x^2 - 9 = (x + 3)(x - 3)$ and $x^2 + x - 6 = (x + 3)(x - 2)$
2. Cancel $(x + 3)$: $\dfrac{x - 3}{x - 2}$

Worked solution

1. $2x^2 + 7x + 3 = (2x + 1)(x + 3)$ and $x^2 - 9 = (x + 3)(x - 3)$
2. Cancel $(x + 3)$: $\dfrac{2x + 1}{x - 3}$

Worked solution

1. $\dfrac{3(x + 1)}{x(x + 1)} + \dfrac{2x}{x(x + 1)}$
2. $= \dfrac{3x + 3 + 2x}{x(x + 1)} = \dfrac{5x + 3}{x(x + 1)}$

Worked solution

1. $\dfrac{4(x + 1) - 3(x - 2)}{(x - 2)(x + 1)}$
2. $4x + 4 - 3x + 6 = x + 10$
3. $= \dfrac{x + 10}{(x - 2)(x + 1)}$

Worked solution

1. $\dfrac{6x}{3x^2}$
2. Cancel 3 and one $x$: $\dfrac{2}{x}$

Worked solution

1. $3(x + 3) - 2(x - 1) = 24$
2. $3x + 9 - 2x + 2 = 24$
3. $x + 11 = 24$, so $x = 13$

Worked solution

1. $6x - 2(x + 1) = x(x + 1)$
2. $4x - 2 = x^2 + x$
3. $x^2 - 3x + 2 = 0$, so $(x - 1)(x - 2) = 0$
4. $x = 1$ or $x = 2$ (neither makes a denominator zero)

Worked solution

1. $\dfrac{(x + 2)(x - 2)}{(x - 2)(x + 1)} \times \dfrac{x + 1}{x + 2}$
2. Every bracket cancels: $(x + 2)$, $(x - 2)$ and $(x + 1)$.
3. The answer is $1$.