Check Up

  1. Justine simplified the rational expression \(\frac{{4x + 12}}{{2x^2 - 7x - 4}} \div \frac{{x + 3}}{{x - 4}} + \frac{{x - 4}}{{2x + 1}}\). Check her work to see if she did it correctly.  If there are errors, fix them.

    \[ {\color{blue} \begin{align}
     \frac{{4x + 12}}{{2x^2 - 7x - 4}} \div \frac{{x + 3}}{{x - 4}} + \frac{{x - 4}}{{2x + 1}} &= \frac{{4\left( {x + 3} \right)}}{{\left( {2x + 1} \right)\left( {x - 4} \right)}} \div \frac{{x + 3}}{{\cancel{x - 4}}} + \frac{{\cancel{x - 4}}}{{2x + 1}} \\
      &= \frac{{4\left( {x + 3} \right)}}{{\left( {2x + 1} \right)\left( {x - 4} \right)}} \div \frac{{x + 3}}{{2x + 1}} \\
      &= \frac{{4 \cancel{\left( {x + 3} \right)}}}{{ \cancel{\left( {2x + 1} \right)} \left( {x - 4} \right)}}\cdot \frac{{\cancel{2x + 1}}}{{\cancel{x + 3}}} \\
      &= \frac{4}{{x - 4}}, x \ne - \frac{1}{2}, 4 \\
     \end{align}}\]



  2. Simplify and identify any non-permissible values for the rational expression \(\frac{{3 - \frac{9}{d}}}{{d - \frac{9}{d}}}\).