# Solving Radical Inequalities

How to solve radical inequalities (square root inequalities): examples and their solutions.

## Example 1: Solve √2*x* - 3 < 5

Just like solving a radical equation,

first find the range of *x*

from the given inequality.

For an even root (√, ^{4}√, ^{6}√, etc.),

the radicand (the number inside the radical)

cannot be (-).

So the radicand of √2*x* - 3, [2*x* - 3],

cannot be (-).

So 2*x* - 3 ≥ 0.

Solving radical equations

Move -3 to the right side.

Then 2*x* ≥ 3.

Divide both sides by 2.

Then *x* ≥ 3/2.

2 is (+).

So the order of the inequality sign

doesn't change.

Next, solve the radical inequality.

Square both sides.

(√2*x* - 3)^{2} = 2*x* - 3

Square root - Square of a square root

5^{2} = 25

Move -3 to the right side.

Then 2*x* < 28.

Divide both sides by 2.

Then *x* < 14.

*x* ≥ 3/2*x* < 14

Draw these two inequalities on a number line.

Find the intersecting region.

Then 3/2 ≤ *x* < 14.

## Example 2: Solve *x* > √*x* - 2 + 1

Find the range of *x*

from the given inequality.

The radicand of √*x*, [*x*],

cannot be (-).

So *x* ≥ 0.

The radicand of √*x* - 2, [*x* - 2],

cannot be (-).

So *x* - 2 ≥ 0.

Move -2 to the right side.

Then *x* ≥ 2.

*x* ≥ 0*x* ≥ 2

Draw these two inequalities on a number line.

And find the intersection.

Then *x* ≥ 2.

Next, solve the radical inequality.

Square both sides.

(√*x*)^{2} = *x*

(√*x* - 2 + 1)^{2} = (√*x* - 2)^{2} + 2⋅√*x* - 2 ⋅1 + 1^{2}

Square of a sum

√*x*, √*x* - 2, and +1 are all not (-).

So both sides are not (-).

So squaring both sides

doesn't change the order of the inequality sign.

(√*x* - 2)^{2} = *x* - 2

2⋅√*x* - 2 ⋅1 = 2√*x* - 2

1^{2} = 1

Cancel *x* on both sides.

(dark gray)

-2 + 1 = -1

Then 0 > 2√*x* - 2 - 1.

Move -1 to the left side.

Then 1 > 2√*x* - 2.

Switch both sides.

Divide both sides by 2.

Then √*x* - 2 < 1/2.

Square both sides.

Then *x* - 2 < 1/4.

√*x* - 2 and 1/2 are not (-).

So both sides are not (-)

So the order of the inequality sign

doesn't change.

Move -2 to the right side.

Then *x* < 1/4 + 2.

2 = 2⋅(4/4)

2⋅(4/4) = 8/4

1/4 + 8/4 = 9/4.

So *x* < 9/4.

*x* ≥ 2*x* < 9/4

Draw these two inequalities on a number line.

Find the intersecting region.

Then 2 ≤ *x* < 9/4.