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Adding $\sqrt{x}$ to both sides, we get $$\sqrt{x+7}=\sqrt{x}+\sqrt{3}.$$Then, squaring both sides gives $$x+7=x+3+2\sqrt{3x},$$or $$4=2\sqrt{3x}.$$Hence, $2=\sqrt{3x},$ so $4=3x$ and $x=\boxed{\frac{4}{3}}.$