Solving Radical Inequalities

Solving Radical Inequalities

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

Example 1: Solve √2x - 3 < 5

Solve the given inequality. Square root (2x - 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 √2x - 3, [2x - 3],
cannot be (-).

So 2x - 3 ≥ 0.

Solving radical equations

Move -3 to the right side.

Then 2x ≥ 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.

(√2x - 3)2 = 2x - 3

Square root - Square of a square root

52 = 25

Move -3 to the right side.

Then 2x < 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

Solve the given inequality. Square root x > square root (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 + 12

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

12 = 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.