Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$

Solve for $$x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}$$

Show that the square of $$\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}$$ is a rational number.

Solve for $$x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.$$