Irrational inequalities and equivalent systems

In the Russian programme irrational inequalities are met in grades 10-11 at the advanced level; the basic course does not teach them. The method is to replace the inequality with an equivalent system or set of systems.

The inequality √f < g is equivalent to a system of three conditions: f ≥ 0 (the domain, ODZ), g > 0 and f < g². The inequality √f > g splits into two systems: either g < 0 and f ≥ 0, or g ≥ 0 and f > g². The solutions are joined together (a set).

If the right-hand side is a number it is easier: √f < 2 is equivalent to 0 ≤ f < 4. If the right-hand side is negative and the domain holds, then √f > −3 is always true.

Squaring gives a consequence equation: extraneous roots may appear, so a check is needed. An equivalent step needs no check, but you must keep the domain and the sign of the right-hand side.

Draw the solution on the number line. When a set is needed you can draw two pieces.