Answer
No solution.
Work Step by Step
We are given:
$\sqrt{x+7}+3=\sqrt{x-4}$
We square both sides:
$(\sqrt{x+7}+3)^{2}=(\sqrt{x-4})^{2}$
And distribute:
$(\sqrt{x+7}+3)(\sqrt{x+7}+3)=(\sqrt{x-4})(\sqrt{x-4})$
$(x+7)+6\sqrt{x+7}+9=x-4$
And combine like terms:
$6\sqrt{x+7}+16+4=x-x$
$6\sqrt{x+7}=-20$
$3\sqrt{x+7}=-10$
We square both sides again:
$(3\sqrt{x+7})^{2}=(-10)^{2}$
$9(x+7)=100$
And distribute:
$9x+63=100$
$9x=37$
And solve for $x$:
$x=\frac{37}{9}$
However, if we plug in $\frac{37}{9}$ into the original equation, it does not work:
$\sqrt{\frac{37}{9}+7}+3=\sqrt{\frac{37}{9}-4}$
$\sqrt{\frac{37}{9}+\frac{63}{9}}+3=\sqrt{\frac{37}{9}-\frac{36}{9}}$
$\sqrt{\frac{100}{9}}+3=\sqrt{\frac{1}{9}}$
$\frac{\sqrt{100}}{\sqrt{9}}+3=\frac{\sqrt{1}}{\sqrt{9}}$
$\frac{10}{3}+3=\frac{1}{3}$
$\frac{10}{3}+\frac{9}{3}=\frac{1}{3}$
$\frac{19}{3}=\frac{1}{3}$
$19=1$
Since we got a false statement, the solution $\frac{37}{9}$ does not work in the original equation. Thus there is no solution.