Irrational inequalities

The concept of the irrational inequality

Definition: Irrational inequality an inequality that contains a variable under the sign of root -degree.

Solution of irrational equations

Interval methods for the solution of irrational inequalities

1. Find DHS inequality.
2. Find the zeros of the function
3. To cancel the zeros of the function at DHS and find the sign function on each of the intervals which split the DHS .

Example 1:

Rozwarte equation:

Solution: the Given inequality is equivalent to inequality

We denote

DHS: i.e.

Zeros: put into the square left and right side

- the root is outside the root.

Answer:

Equivalent conversion

1. In the presentation of both parts of the inequality to an odd degree (with preservation of the inequality sign) we get the inequality, tantamount given.

Example 2:

Rozwarte equation:

Solution: DHS:

The given inequality is equivalent to the inequalities:

Answer:

2. If both sides of an inequality newmn, when you lift both parts of the inequality to the pair of degrees (preserving the sign of the inequality) we get the inequality, tantamount given.

Example 3 :

Rozwarte equation:

Solution: DHS:

Both parts of the given inequality newmn, therefore, it is equivalent to the inequalities:

Given DHS obtained .

Answer:

3. If DHS inequality given some part of the inequality can acquire both positive and integral values, before bringing both parts of the inequality to the pair anymore, these cases should be considered separately.

Example 4 :

Rozwarte equation:

Solution: the Given inequality is equivalent to the set of systems:

or

Then or

Rozwadowski inequality have

Given the inequalities , we obtain the solution of the first system . The solution of the second system: Obanyaki of these solutions, we get the answer.

Answer: