Example 7
Completion requirements
| Example 7 |
Simplify the rational expression \(\frac{{2\left( {x + 1} \right)}}{{5\left( {x - 3} \right)}} \div \frac{{3\left( {x + 1} \right)}}{{25\left( {x + 3} \right)}}\). Identify any non-permissible values.
Step 1: Identify the NPVs.
Step 2: Multiply by the reciprocal.
Step 3: Identify any new NPVs.
\(\begin{align}
x + 1 &\ne 0 \\
x &\ne - 1 \\
\end{align}\)
Step 4: Simplify.
\(\begin{align}
x - 3 &\ne 0 \\
x &\ne 3 \\
\end{align}\)
x - 3 &\ne 0 \\
x &\ne 3 \\
\end{align}\)
\(\begin{align}
x + 3 &\ne 0 \\
x &\ne - 3 \\
\end{align}\)
x + 3 &\ne 0 \\
x &\ne - 3 \\
\end{align}\)
Step 2: Multiply by the reciprocal.
\[\frac{{2\left( {x + 1} \right)}}{{5\left( {x - 3} \right)}} \div
\frac{{3\left( {x + 1} \right)}}{{25\left( {x + 3} \right)}} =
\frac{{2\left( {x + 1} \right)}}{{5\left( {x - 3} \right)}}\cdot
\frac{{25\left( {x + 3} \right)}}{{3\left( {x + 1} \right)}}\]
Step 3: Identify any new NPVs.
\(\begin{align}
x + 1 &\ne 0 \\
x &\ne - 1 \\
\end{align}\)
Step 4: Simplify.
\[\begin{align}\frac{{2\left( {x + 1} \right)}}{{5\left( {x - 3} \right)}} \div
\frac{{3\left( {x + 1} \right)}}{{25\left( {x + 3} \right)}} &=
\frac{{2{\color{red} \cancel {\color{#444} {{\left( {x + 1} \right)}}}}}}{{{\color{red} \cancel {\color{#444} 5}}\left( {x - 3} \right)}}\cdot
\frac{{{\color{red} \cancel {\color{#444}{25}}^5}\left( {x + 3} \right)}}{{3 {\color{red} \cancel {\color{#444}\left( {x + 1} \right)}}}} \\
&= \frac{2}{x - 3} \cdot \frac{5(x - 3)}{3} \\
&= \frac{10(x + 3)}{3 ( x - 3)} \cdot x \ne \pm3, -1\end{align}\]
&= \frac{2}{x - 3} \cdot \frac{5(x - 3)}{3} \\
&= \frac{10(x + 3)}{3 ( x - 3)} \cdot x \ne \pm3, -1\end{align}\]