Solving Radical Inequalities

Solving radical inequalities is similar to solving rational equations, but there is one extra step since we must make sure the radical is a real number, i.e. the radicand must be greater than or equal to zero.

Example 1 Solve Since the radical must be a real number, must be greater than or equal to zero.

Square both sides Subtract 2 from both sides We know that both and must be true.

Check a value of in the original inequality. The solution is all real numbers such that. [Note: this takes care of also]

Example 2 Solve Since each radical must be a real number, for the first radical so and for the other radical. makes both radicals real.

Solve Isolate one radical Square both sides Simplify (continued on next slide)

(continued from previous slide) Isolate one variable Square both sides Simplify and solve Noting also that, the solution is, approximately.

Check your answer by substituting a value for x in the original inequality.