# Radical Inequality

How to solve a radical inequality: 2 examples and their solutions.

## Example 1

### Example

### Solution

Just like solving a radical equation,

first find the condition from the x.

See √2x - 3.

The number inside the square root

should be (+) (or 0).

So 2x - 3 ≥ 0.

Move -3 to the right side.

Then 2x ≥ 3.

Divide both sides by 2.

Then x ≥ 3/2.

So x should satisfy

x ≥ 3/2.

This is the condition for x.

Next, solve the given radical inequality

√2x - 3 < 5.

Square both sides.

Then

2x - 3 < 5^{2}.

√2x - 3 is (+).

5 is also (+).

Squaring both sides means

multiplying (+) numbers on both sides.

So squaring both sides

does not change

the order of the inequality sign.

Linear Inequality (One Variable)

5^{2} = 25

Move -3 to the right side.

Then 2x < 28.

Divide both sides by 2.

Then x < 14.

The x condition was

x ≥ 3/2.

And you got

x < 14.

Draw these two inequalities

on a number line.

x should satisfy both inequalities.

So color the common region.

The colored region is

3/2 ≤ x < 14.

So

3/2 ≤ x < 14

is the answer.

## Example 2

### Example

### Solution

First find the conditions from each x.

See √x.

The number inside the square root

should be (+) (or 0).

So x ≥ 0.

Next, see √x - 2.

The number inside the square root

should be (+) (or 0).

So x - 2 ≥ 0.

Move -2 to the right side.

Then x ≥ 2.

x ≥ 0

x ≥ 2

Draw the inequalities on a number line.

x should satisfy both conditions.

So color the common region.

The common region is

x ≥ 2.

So x should satisfy

x ≥ 2.

This is the condition for x.

Next, solve the given radical inequality

√x > √x - 2 + 1.

Square both sides.

The left side is

(√x)^{2} = x.

The right side is

(√x - 2 + 1)^{2}

= (x - 2) + 2⋅√x - 2⋅1 + 1^{2}.

Square of a Sum: (a + b)^{2}

The left side, √x, is (+).

The right side, √x - 2 + 1, is also (+).

So squaring both sides

does not change

the order of the inequality sign.

Cancel the x on both sides.

+2⋅√x - 2⋅1

= +2√x - 2

+1^{2} = +1

-2 + 1 = -1

Move -1 to the left side.

Switch both sides.

Don't forget to change the inequality sign:

> → <.

Divide both sides by 2.

Square both sides.

Then the left side is

(√x - 2)^{2} = x - 2.

The right side is

(1/2)^{2} = 1/4.

Both √x - 2 and 1/2 are (+).

So squaring both sides

does not change

the order of the inequality sign.

Move -2 to the right side.

+2 = +2⋅4/4

+2⋅4/4 = +8/4

1/4 + 8/4 = 9/4

So x < 9/4.

The x condition was

x ≥ 2.

And you got

x < 9/4.

Draw these two inequalities

on a number line.

x should satisfy both inequalities.

So color the common region.

The colored region is

2 ≤ x < 9/4.

So

2 ≤ x < 9/4

is the answer.