EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS

Note:


Problem2.75         $\sqrt[11]{\displaystyle \frac{3x+11}{2}}+5=4$

Answer:         $x=-\displaystyle \frac{13}{3}$

Solution:

Isolate the radical term.

\begin{eqnarray*}\sqrt[11]{\displaystyle \frac{3x+11}{2}} &=&-1 \\
&& \\
&&
\end{eqnarray*}


Raise both sides of the equation to the 11th power.


\begin{eqnarray*}\left( \sqrt[11]{\displaystyle \frac{3x+11}{2}}\right) ^{11} &=...
...\\
&& \\
x &=&-\displaystyle \frac{13}{3} \\
&& \\
&& \\
&&
\end{eqnarray*}


Check your answer by substituting $x=-\displaystyle \frac{13}{3}$ in the original equation. If the left side of the original equation equals the right side of the original equation after the substitution, the answer $x=-\displaystyle \frac{13}{3}$

Left Side: $\qquad \sqrt[11]{\displaystyle \frac{3\left( -\displaystyle \frac{13}{3}\right) +11}{2}}
+5=-1+5=4$

Right Side$\qquad 4$

You can also check the answer by graphing the equation:


\begin{eqnarray*}y &=&\sqrt[11]{\displaystyle \frac{3x+11}{2}}+1 \\
&& \\
&&
\end{eqnarray*}


The graph represents the right side of the original equation minus the left side of the original equation.. Note that the x-intercept on the graph is located at $-\displaystyle \frac{13}{3}$, this confirms that $-\displaystyle \frac{13}{3}$ is our solution.


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Author:Nancy Marcus

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