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 < 52.

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)

52 = 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 + 12.

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

+12 = +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.