Solving Radical Inequalities

Solving Radical Inequalities

How to solve radical inequalities: examples and their solutions.

Example 1

Solve the given inequality. Square root (2x - 3) < 5.

First, find the range of x.

2x - 3 is the radicand of the square root.
So 2x - 3 ≥ 0.

x ≥ 3/2

To remove the radical sign,
square both sides.

The order of the inequality sign doesn't change
because both sides (√2x - 3, 5) are (+).

Then x < 14.

Draw x ≥ 3/2 and x < 14 on a number line.

Graphing linear inequalites on a number line

Find the intersecting region:
3/2 ≤ x < 14.

Example 2

Solve the given inequality. Square root x > square root (x - 2) + 1.

Find the ranges of x.

x is the radicands of the square root.
So x ≥ 0.

x - 2 is also the radicands of the square root.
So x - 2 ≥ 0.
x ≥ 2

To remove the radical sign of √x,
square both sides.

Square of a sum

To remove the radical sign of √x - 2,
leave only √x - 2 on the left side,
and square both sides again.

Draw x ≥ 0, x ≥ 2, and x < 9/4 on a number line.

Graphing linear inequalites on a number line

Find the intersecting region:
2 ≤ x < 9/4.