Irrational inequalities

An inequality with the unknown under a root sign is an irrational inequality. Since the solution set is infinite, an answer cannot be checked by substitution, so every transformation must be equivalent. Raising both sides to an even power gives an equivalent inequality only when both sides are nonnegative. Rules: √A < B ⇔ {A < B², A ≥ 0, B ≥ 0}; √A > B ⇔ either {B < 0, A ≥ 0} or {B ≥ 0, A > B²} (the union of two cases); √A > √B ⇔ {B ≥ 0, A > B}. Also √A ≤ B ⇔ {A ≥ 0, B ≥ 0, A ≤ B²}. The first inequality comes from squaring, the second from the existence of the root, the third from the right to square (B ≥ 0). If B < 0, then √A < B has no solution, while √A > B holds for every x with A ≥ 0.