Simplify Rational Expression

This example is about simplifying rational sxpressions.

Example 1


$\dfrac{1}{3+\dfrac{1}{x+3}}=$


  1. $\dfrac{3x+10}{x+3}$

  2. $3+\dfrac{1}{x+3}$

  3. $\dfrac{x+3}{3x+10}$

  4. $\dfrac{x+3}{3}$

  5. $\dfrac{1}{x+6}$

Solution

This example is about simplifying rational expressions.

  1. The 5 possible answers are simple rational expressions and therefore we need to rewrite the given expression as a simple rational expression in order to be able to compare it to the 5 answers. Let us add the two terms in the denominator $3+\dfrac{1}{x+3}$ as follows

    $3+\dfrac{1}{x+3}=\dfrac{3(x+3)}{x+3}+\dfrac{1}{x+3}=\dfrac{3x+10}{x+3}$

    The given expression is equivalent to

    $\dfrac{1}{\dfrac{3x+10}{x+3}}$

    Rewrite $1$ in the numerator as a fraction

    $= \dfrac{\dfrac{1}{1}}{\dfrac{3x+10}{x+3}}$

    To divide two rational expressions, we multiply the first by the reciprocal of the second. Hence

    $=\dfrac{1}{1} \times \dfrac{x+3}{3x+10}$

    $=\dfrac{x+3}{3x+10}$

    Answer C