Square Root Inequality Calculator

Square root inequalities involve expressions like √x > a or √x < b. Solving these requires careful consideration of the domain and squaring both sides to eliminate the square root. This calculator helps you solve such inequalities accurately.

How to Solve Square Root Inequalities

Solving square root inequalities follows a specific set of rules to ensure the solution is accurate. Here's what you need to know:

Key Rule: The expression under the square root must be non-negative (x ≥ 0) because the square root of a negative number is not a real number.

Step-by-Step Solution

Identify the domain of the inequality (x ≥ 0).
Square both sides of the inequality to eliminate the square root.
Solve the resulting quadratic inequality.
Combine the solutions with the domain to find the final solution set.

If √x > a, then: 1. x ≥ 0 2. x > a² 3. Solution: x > a²

Worked Examples

Example 1: √x > 3

Domain: x ≥ 0
Square both sides: x > 9
Combine with domain: x > 9

Example 2: √x < 2

Domain: x ≥ 0
Square both sides: x < 4
Combine with domain: 0 ≤ x < 4

Common Mistakes to Avoid

Forgetting to consider the domain (x ≥ 0).
Squaring both sides without considering the inequality sign.
Not checking the solution against the original inequality.

Frequently Asked Questions

What is the domain of a square root inequality?: The domain is all real numbers x such that x ≥ 0, because the square root of a negative number is not a real number.
Can I square both sides of a square root inequality?: Yes, but you must ensure the inequality sign remains correct. The direction of the inequality sign changes if you multiply or divide by a negative number.
What if the inequality has a negative number under the square root?: The inequality has no real solutions because the square root of a negative number is not a real number.
How do I solve √x > -2?: This inequality has no real solutions because the square root function always returns a non-negative value, and -2 is negative.