$$\begin{array}{r r c c c l c r c c c l} \left| x^2 - 64 \right| < \frac{3}{1000} \implies & -\frac{3}{1000} & < & x^2 - 64 & < & \frac3{1000} \\ & 64 - \frac3{1000} & < & x^2 & < & 64 + \frac3{1000} \\ & \frac{63997}{1000} & < & x^2 & < & \frac{64003}{1000} \\ & -\sqrt{\frac{64003}{1000}} & < & x & < & -\sqrt{\frac{63997}{1000}} & \text{or} & \sqrt{\frac{63997}{1000}} & < & x & < & \sqrt{\frac{64003}{1000}} \end{array}$$
Since we need $x \approx 8$ we take
$$\begin{array}{r c c c l l} \sqrt{\frac{63997}{1000}} & < & x & < & \sqrt{\frac{64003}{1000}} \\ \sqrt{\frac{63997}{1000}} - 8 & < & x - 8 & < & \sqrt{\frac{64003}{1000}} - 8 \\ -0.0001875 & < & x-8 & < & 0.0001875 & \implies |x-8| < 0.000187 \end{array}$$