Radical Functions: Solving Radical Equations and Inequalities

Solving Radical Inequalities

Now that you have a good grasp of how to solve radical equations, let's move on to radical inequalities. Let's work through an example as we learn about solving radical inequalities.

Solve square root of the quantity x minus one is less than two.

One of the first things that you need to remember is that anything inside the radical sign of a square root function must be greater than or equal to zero.

In this case x – 1 ≥ 0 so x ≥ 1. If this is not true then our whole inequality is false.

The next step is to solve the original inequality.

Square root of the quantity x minus one is less than two. Square root of the quantity x minus one. Squared is less that two squared, x minus one is less than four. x is less than five

The next step is to graph these two statements on a number line. Remember, these two statements are linked by an "and". Both must be true in order for the inequality to be true.

x ≥ 1 and x < 5

x is greater than or equal to one and less than 5, closed dot at one and open dot at five, shaded in between

The final answer is x ≥ 1 and x < 5 or 1 ≤ x < 5.

Let's analyze a table of values for this function:

x

f of x equals square root of quantity x minus one

0

Undefined

1

0

2

1

3

1.4

4

1.7

5

2

6

2.2

The values of x = 1, x = 2, x = 3, and x = 4 all have f(x) values that are less than 2, which is the inequality that must hold true for our problem. When we get to x = 5 and all x values larger that
x = 5, the f(x) value is greater than or equal to 2. Analyzing this graph can show you another way of reaching the final solution of x ≥ 1 and x < 5 or 1 ≤ x < 5.