# Section A.4 – Rational Expressions

Section Details.
• Definition and properties of rational expressions
• Simplifying rational expressions
• Operations with rational expressions
• Simplifying compound fractions

### Practice Problems

Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Perform the operations and simplify: $$\dfrac{x^2-3x-10}{2x^2-9x-5}\div \dfrac{x^2-2x-8}{2x^2-9x+4}$$

Answer: $$\dfrac{2x-1}{2x+1}$$

Solution: \begin{align} \dfrac{x^2-3x-10}{2x^2-9x-5}\div \dfrac{x^2-2x-8}{2x^2-9x+4} &=\dfrac{(x-5)(x+2)}{(2x+1)(x-5)}\div\dfrac{(x-4)(x+2)}{(2x-1)(x-4)}\\ &=\dfrac{x+2}{2x+1}\cdot\dfrac{2x-1}{x+2}\\ &=\dfrac{2x-1}{2x+1} \end{align}

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2. Perform the operations and simplify: $$\dfrac{x+1}{x^2-4x+4}-\dfrac{x-3}{x^2-4}$$

Answer: $$\dfrac{4(2x-1)}{(x-2)^2(x+2)}$$

Solution: \begin{align} \dfrac{x+1}{x^2-4x+4}-\dfrac{x-3}{x^2-4} &=\dfrac{x+1}{(x-2)(x-2)}-\dfrac{x-3}{(x-2)(x+2)}\\ &=\dfrac{x+1}{(x-2)(x-2)}\cdot\dfrac{x+2}{x+2}-\dfrac{x-3}{(x-2)(x+2)}\cdot\dfrac{x-2}{x-2}\\ &=\dfrac{x^2+3x+2}{(x-2)(x-2)(x+2}-\dfrac{x^2-5x+6}{(x-2)(x-2)(x+2)}\\ &=\dfrac{8x-4}{(x-2)(x-2)(x+2)}\\ &=\dfrac{4(2x-1)}{(x-2)^2(x+2)} \end{align}

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3. Perform the operations and simplify: $$\dfrac{\displaystyle\frac{2}{x}+\displaystyle\frac{1}{3x^2}}{\displaystyle\frac{4}{x}-1}$$

Answer: $$\dfrac{6x+1}{3x(4-x)}$$

Solution: We can clear out the complex fraction by multiplying top and bottom by the ENTIRE fraction's common denominator.
\begin{align} \dfrac{\displaystyle\frac{2}{x}+\displaystyle\frac{1}{3x^2}}{\displaystyle\frac{4}{x}-1} &=\dfrac{\dfrac{2}{x}+\dfrac{1}{3x^2}}{\dfrac{4}{x}-1}\cdot\dfrac{3x^2}{3x^2}\\ &=\dfrac{6x+1}{12x-3x^2}\\ &=\dfrac{6x+1}{3x(4-x)} \end{align}

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4. Perform the multiplication and simplify: $$\displaystyle \frac{t^2-t-6}{t^2+6t+9}\cdot\frac{t+3}{t^2-4}$$

$$\dfrac{t-3}{(t+3)(t-2)}$$

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