Example 3
Completion requirements
| Example 3 |
Simplify the expression \(\left[ {\frac{{x - 2}}{{5x + 1}} - \frac{{x - 2}}{{3x - 2}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}}\). Identify any non-permissible values.
Step 1: Identify the NPVs.
Step 2: Determine which operation to perform first, and simplify.
According to BEDMAS, the subtraction in brackets must be done first.
Notice the use of grouping (removing a factor of \(x - 2\) from each term in the numerator) to help in the simplification process.
Step 3: Perform the last operation, and simplify.
Division is the last operation.
Pause, and identify any new NPVs.
\(\begin{align}
x - 2 &\ne 0 \\
x &\ne 2 \\
\end{align}\)
Continue simplifying.
\(\begin{align}
5x + 1 &\ne 0 \\
x &\ne - \frac{1}{5} \\ \end{align}\)
5x + 1 &\ne 0 \\
x &\ne - \frac{1}{5} \\ \end{align}\)
\(\begin{align}
3x - 2 &\ne 0 \\
x \ne &\frac{2}{3} \\
\end{align}\)
3x - 2 &\ne 0 \\
x \ne &\frac{2}{3} \\
\end{align}\)
\(\begin{align}
x + 1 &\ne 0 \\
x &\ne - 1 \\
\end{align}\)
x + 1 &\ne 0 \\
x &\ne - 1 \\
\end{align}\)
Step 2: Determine which operation to perform first, and simplify.
According to BEDMAS, the subtraction in brackets must be done first.
\[\begin{align}
\left[ {\color{red} {\frac{{\left( {x - 2} \right)}}{{\left( {5x + 1} \right)}} - \frac{{\left( {x - 2} \right)}}{{\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} &= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( {3x - 2} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} - \frac{{\left( {x - 2} \right)\left( {5x + 1} \right)}}{{\left( {3x - 2} \right)\left( {5x + 1} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( {3x - 2} \right) - \left( {x - 2} \right)\left( {5x + 1} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left[ {3x - 2 - \left( {5x + 1} \right)} \right]}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( {3x - 2 - 5x - 1} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
\end{align}\]
\left[ {\color{red} {\frac{{\left( {x - 2} \right)}}{{\left( {5x + 1} \right)}} - \frac{{\left( {x - 2} \right)}}{{\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} &= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( {3x - 2} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} - \frac{{\left( {x - 2} \right)\left( {5x + 1} \right)}}{{\left( {3x - 2} \right)\left( {5x + 1} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( {3x - 2} \right) - \left( {x - 2} \right)\left( {5x + 1} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left[ {3x - 2 - \left( {5x + 1} \right)} \right]}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( {3x - 2 - 5x - 1} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
&= \left[ {\color{red}{\frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} \\
\end{align}\]
Notice the use of grouping (removing a factor of \(x - 2\) from each term in the numerator) to help in the simplification process.
Step 3: Perform the last operation, and simplify.
Division is the last operation.
\[\begin{align}
\left[ {\frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} &= \frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} \cdot \frac{{x + 1}}{{x^2 - x - 2}} \\
&= \frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}\cdot \frac{{x + 1}}{{\left( {x - 2} \right)\left( {x + 1} \right)}} \\
\end{align}\]
\left[ {\frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}} \right] \div \frac{{x^2 - x - 2}}{{x + 1}} &= \frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} \cdot \frac{{x + 1}}{{x^2 - x - 2}} \\
&= \frac{{\left( {x - 2} \right)\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}\cdot \frac{{x + 1}}{{\left( {x - 2} \right)\left( {x + 1} \right)}} \\
\end{align}\]
Pause, and identify any new NPVs.
\(\begin{align}
x - 2 &\ne 0 \\
x &\ne 2 \\
\end{align}\)
Continue simplifying.
\[\begin{align}
\frac{{{\color{red} \cancel {\color{#444}(x - 2)}}\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} \cdot \frac{{{\color{red} \cancel {\color{#444}(x + 1)}}}}{{{\color{red} \cancel {\color{#444}(x - 2) } \cancel {\color{#444}(x + 1)}}}} &= \frac{{ - \left( {2x + 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} \\
&= - \frac{{2x + 3}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}, \thinspace x \ne - 1, - \frac{1}{5},\frac{2}{3}, \thinspace 2 \\
\end{align}\]
\frac{{{\color{red} \cancel {\color{#444}(x - 2)}}\left( { - 2x - 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} \cdot \frac{{{\color{red} \cancel {\color{#444}(x + 1)}}}}{{{\color{red} \cancel {\color{#444}(x - 2) } \cancel {\color{#444}(x + 1)}}}} &= \frac{{ - \left( {2x + 3} \right)}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}} \\
&= - \frac{{2x + 3}}{{\left( {5x + 1} \right)\left( {3x - 2} \right)}}, \thinspace x \ne - 1, - \frac{1}{5},\frac{2}{3}, \thinspace 2 \\
\end{align}\]