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



    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V16

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



    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V17

  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.
    \[\begin{align}
    \dfrac{\displaystyle\frac{2}{x}+\displaystyle\frac{1}{3x^2}}{\displaystyle\frac{4}{x}-1}
    &=\dfrac{\dfrac{2}{x}+\dfrac{1}{3x^2}}{\dfrac{4}{x}-1}\cdot\dfrac{3x^2}{3x^2}\\
    &=\dfrac{6x+1}{12x-3x^2}\\
    &=\dfrac{6x+1}{3x(4-x)}
    \end{align}\]



    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V18

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

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



    To see the full video page and find related videos, click the following link.
    WIR8 20B M150 V04