Fluency exercise

## Solutions

#### Expressions collection

1. $\dfrac{4}{x} - \dfrac{3}{x-2}$

2. $\dfrac{4x}{5} + \dfrac{8x}{15} - \dfrac{1}{2}$

3. $x - \dfrac{5}{x -2} - \dfrac{5}{x+2}$

4. $\dfrac{x}{x+1} + \dfrac{2x}{x-3} + \dfrac{x^3 - 2x^2 - 11x}{(x-3)(x+1)}$

#### Equations collection

1. $\dfrac{3}{x+3} - \dfrac{1}{x-4} + \dfrac{57}{(x+3)(x-4)} = 1$

2. $\dfrac{4x}{5} - \dfrac{52}{15} = \dfrac{8}{3x}$

3. $\dfrac{3x}{4} +\dfrac{2x^2}{5} = \dfrac{x}{2}$

4. $\dfrac{x^2 - 2x}{x - 4} = \dfrac{2x}{x - 4}$

#### Inequalities collection

1. $\dfrac{3(x-1)}{4} < \dfrac{5x-1}{2}$

2. $\dfrac{4x-3}{2} - \dfrac{2x+2}{5} ≥0$

3. $\dfrac{5}{3} - \dfrac{1}{x} < \dfrac{3}{x}$

4. $\dfrac{5x+6}{x-2} ≤ 2$

Now you have finished try to think about the following questions:

• What varies in your approach to each of the collections, for example, between simplifying expressions and solving equations?

• Is cancelling by a common factor of 2 different from cancelling by a common factor of $x-2$ when dealing with expressions/equations/inequalities?

• What impact can multiplying or dividing by $x$ (or a factor such as $x+3$) have?