um math prep: s8.3 - solving radical ineq

We can eliminate radicals by taking both sides of an inequality to the same power (recall that the inequality isn't changed if both sides are positive and we square them). However, as when we solve equations we must check our answers for extraneous roots.

Examples
Solve each inequality:

The expression is not defined unless , so we can limit the x values that we need to consider to . Also, , so we can square both sides of the inequality to give

However, we are only considering values of x which are 6 or larger, so the solution is
Here the fact that the radical is only defined for x less than or equal to 2 does not affect our solution because any number less than -7 is clearly also less than 2. The solution is
x < -7

Practice

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get a new version of the practice problem by clicking the "Refresh" button at the bottom of the problem.

Solving Radical Inequalities