Answer
No solution
Work Step by Step
Given\begin{equation}
\sqrt{5 x+4}-\sqrt{5 x-7}=-9.
\end{equation} Start by rearranging the equation and squaring both sides to eliminate the radical sign on the left hand side. Rearrange the equation again and then square both sides to eliminate the radical sign on the right hand side and then solve for $x$.
\begin{equation}
\begin{aligned}
\sqrt{5 x+4}-\sqrt{5 x-7}&=-9\\
\sqrt{5 x+4}&=\sqrt{5 x-7}-9\\
\left(\sqrt{5 x+4}\right)^2&= \left( \sqrt{5 x-7}-9\right)^2\\
5 x+4& = 5x-7-18\sqrt{5x-7}+81 \\
-70& =-18\sqrt{5 x-7}\\
\left( 35\right)^2 & = \left( 9\sqrt{5 x-7}\right)^2\\
1225& =405x-567\\
1792& =405x\\
\frac{1792}{405}&= x.
\end{aligned}
\end{equation} Check.
\begin{equation}
\begin{aligned}
\sqrt{5 \cdot \frac{1792}{405}+4}-\sqrt{5 \cdot \frac{1792}{405}-7}&\stackrel{?}{=}-9 \\
1.22& =-9\quad \textbf{False}\\
\end{aligned}
\end{equation} The given equation has no solution.
