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  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.

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

    The first error made by Justine was simplifying the addition as a multiplication. According to BEDMAS, addition should be done after multiplication/division, and secondly, addition requires finding a common denominator before simplifying.

    Another error was when multiplying by the reciprocal; Justine should have added the NPV of \(-3\). The correct solution is

    \[\begin{align}
     \frac{{4x + 12}}{{2x^2 - 7x - 4}} \div \frac{{x + 3}}{{x - 4}} + \frac{{x - 4}}{{2x + 1}} &= \frac{{4\left( {x + 3} \right)}}{{\left( {2x + 1} \right)\left( {x - 4} \right)}} \div \frac{{x + 3}}{{x - 4}} + \frac{{x - 4}}{{2x + 1}} \\
      &= \frac{{ 4{\color{red}\cancel {\color{#444}{\left( {x + 3} \right)}}}}}{{\left( {2x + 1} \right){\color{red}\cancel {\color{#444}{\left( {x - 4} \right)}}}}} \cdot \frac{{{\color{red}\cancel {\color{#444}{x - 4}}}}}{{{\color{red}\cancel {\color{#444}{x + 3}}}}} + \frac{{x - 4}}{{2x + 1}} \\
      &= \frac{4}{{2x + 1}} + \frac{{x - 4}}{{2x + 1}} \\
      &= \frac{{4 + x - 4}}{{2x + 1}} \\
      &= \frac{x}{{2x + 1}}, x \ne - 3, - \frac{1}{2}, 4 \\
     \end{align}\]

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

    \[\begin{align}
     \frac{{3 - \frac{9}{d}}}{{d - \frac{9}{d}}} &= \left( {3 - \frac{9}{d}} \right) \div \left( {d - \frac{9}{d}} \right) \\
      &= \left( {\frac{{3d}}{d} - \frac{9}{d}} \right) \div \left( {\frac{{d^2}}{d} - \frac{9}{d}} \right) \\
      &= \left( {\frac{{3d - 9}}{d}} \right) \div \left( {\frac{{d^2 - 9}}{d}} \right) \\
      &= \frac{{3d - 9}}{{\color{red}\cancel{\color{#444}{d}}}} \cdot \frac{{\color{red}\cancel {\color{#444}{d}}}}{{d^2 - 9}} \\
      &= \frac{{3 {\color{red}\cancel {\color{#444}{\left( {d - 3} \right)}}}}}{{{\color{red}\cancel {\color{#444}{\left( {d - 3} \right)}}}\left( {d + 3} \right)}} \\
      &= \frac{3}{{d + 3}}, d \ne \pm 3, 0 \\
     \end{align}\]




 For further information about multiplying and dividing rational expressions, see pp. 323 to 326 of Pre-Calculus 11.