Solving Radical Inequalities

Instructor: Maria Blojay

Maria has taught College Algebra and has a master's degree in Education Administration.

This lesson will show how to solve radical inequalities using radical definitions and properties. Key terms and radical notes are included to share step by step the method to solve these special types of inequalities. Updated: 12/10/2021

There are some mathematical definitions and properties for radicals that we need to review. These properties will help us to understand how to solve radical inequalities.

Radical Notes

Radical Inequality Definition

The definition of a radical inequality is an inequality that holds a variable expression within it. This means that the variable expression sits underneath the radical, and is called a radicand.

For example:


Definition


Negative Radicands Property

Let's take a look at this property:


Negative Radicand


The radicand cannot have a calculated final value that is negative WHEN the index of the radical is an even number.

Having a negative under the radical when the index is an even number, such as 2, 4, 6, etc. means that there is no solution. This must be checked for each radical inequality problem. We will see some problems using these extra checks.

Removing the Radical Property

We cannot mathematically solve the problem with the radical symbol in it. It must be removed. So don't forget that we have this property that tells us how to do that:


Radical Raised to n


This means that we take the index value and use this same value as an exponent. In doing so, it cancels out the radical symbol and leaves the radicand variable expression by itself.

Let's take a look at an example of this:


Example Removing Radical


If the index is not shown in the radical, it means that the index is equal to two.

Now that we have looked at our radical definitions and properties, we will take a look at three sample problems.

Example 1

Let's take a look at our first problem:


Ex2S1


In simplifying:


Ex2S2


Squaring each side, we get,


Ex2S3


Leaving us with,


Ex2S4


Ex2S5


To check, choose an x-value greater than 40. If we chose a number such as 68 and replaced this selected value for x:


Ex2S6


which is greater than 8. So x >40 is the correct inequality answer.

Example 2

Given our second example:


Ex1S1


To get rid of the radical, we square each side of the inequality:


Ex1S2


We then simplify the inequality and get:


Ex1S3


Ex1S4


Ex1S5


Remember that our radicand can NOT be negative, or another way of saying this is that the radicand must be positive:

To check this ... we get:


Ex1S6


Ex1S7


Ex1S89


Let's check our example with x-values of 3 and 5:


Ex1S10


Ex1S1112


Here we have shown this is a true inequality, 0 is less than 2. Now let's try the x value 5:


Ex1S1315


Yes, we have a true inequality with an x value of 3 which is equal to 2.

The values of x that are 3 and 5 AND all values of x in between 3 and 5 will make the inequality true.

Here, 3 is our lower interval value, and 5 is our upper interval value.

To write all of the values that would solve this inequality concisely, we would write it as a compound inequality.

A compound inequality is written by using the lower and upper interval values with the same inequality symbols in between those values.

So, we would write:


Ex1S16


as our final answer.

Example 3

In our last example, we have a radical expression on each side of the inequality.

Given:


Ex3S1


Squaring each side:


Ex3S2


Leaving us with:


Ex3S3


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