Radical side must exist
For sqrt(f(x)), start with f(x) >= 0.
This domain rule stays active through the entire problem. Even a correct squared inequality is rejected if it includes values outside the radical domain.
Radical Inequality Calculator
Solve square root inequalities step-by-step — and catch the extraneous solutions squaring can create.
solve square root inequalityradical inequality solver with stepsextraneous solution check
Why this page is different
It treats the radical domain f(x) >= 0 as the first required condition, not a footnote.
It blocks the hidden trap where squaring makes false x-values look valid.
It uses a two-graph radar so students can see the extraneous region instead of memorizing a warning.
Calculator
Solve a square root inequality
√
Live expression
Answer
Domain:
· Mode: radical vs constant
1. Check the square-root domain first
The expression under the radical must be nonnegative. This page keeps that restriction before doing any squaring.
2. Square both sides only after checking the right side
The constant is nonnegative, so squaring preserves the comparison for this one-sided radical inequality.
3. Intersect the squared result with the domain
The squared inequality may be broader than the original radical problem, so the domain remains active.
4. Verify boundary and sample values
The calculator checks representative values against the original inequality, not only the squared version.
Extraneous-solution radar
See that red shaded area? Those are x-values that satisfy the squared inequality, but not the original inequality. Always verify after squaring.
Original inequality
compare curves before squaring
square-root sidecomparison side
After squaring
red = extraneous region
radicandright side squared
Case branches
Constant comparison branch
5 is nonnegative
The answer keeps the domain condition and verifies the original inequality.
Verification samples
x = -3works
original: 0 vs 5 · squared: 0 vs 25
x = -2.75works
original: 0.5 vs 5 · squared: 0.25 vs 25
x = -2works
original: 1 vs 5 · squared: 1 vs 25
x = -1works
original: 1.4142 vs 5 · squared: 2 vs 25
Breakdown
Step-by-step radical inequality logic
01
Enter a square root inequality such as sqrt(x + 3) < 5 or sqrt(x + 3) < x - 1.
02
Review the domain check first: the expression under the radical must be greater than or equal to 0.
03
Follow the sign-case branch before squaring, then compare the final interval with the original inequality check.
Decision Rules
When squaring is safe, conditional, or misleading
Less-than needs a positive comparison side
For sqrt(f(x)) < g(x), also require g(x) > 0.
A nonnegative square root cannot be less than a negative number, so the non-radical side must be positive before squaring is valid.
Greater-than can split into cases
For sqrt(f(x)) > g(x), values with g(x) < 0 often pass automatically inside the domain.
When the right side is negative, any defined square root is greater than it. When the right side is nonnegative, square both sides and verify.
Common Mistakes
The checks that prevent extraneous answers
Squaring before checking the domain
Write the radicand condition first and intersect it with every later interval.
Keeping values where the comparison side is negative
For strict less-than comparisons, require the right side to be positive before the square step.
Trusting the squared inequality without verification
Substitute test values back into the original radical inequality, especially around new endpoints.
Examples
Common radical inequality types
Radical vs positive constant
Check x + 3 >= 0, square against 5, then intersect x >= -3 with x < 22.
Result: -3 <= x < 22
Radical vs negative constant
A square root is never negative, so this is a valid no-solution outcome, not an error.
Result: No Solution
Expression side with extraneous check
Require x - 1 > 0 before squaring. Squaring alone admits extra values, so the final verified answer is x > (3 + sqrt(17)) / 2.
Result: x > 3.5616
FAQ
Frequently asked questions
Why do you need to check the domain when solving radical inequalities?
Because a square root only exists in the real-number setting when the radicand is at least 0. For sqrt(f(x)), the calculator always starts with f(x) >= 0 before touching the inequality.
Can squaring both sides of an inequality create extraneous solutions?
Yes. Squaring can make negative-side comparisons look true even when the original square-root inequality was false. That is why this calculator verifies sample and boundary values in the original inequality after solving.
What happens when a square root is compared to a negative number?
A square root is always nonnegative. So sqrt(f(x)) < a negative number has no solution, while sqrt(f(x)) > a negative number is true for every x in the radical domain.
How do you solve sqrt(x) < g(x) when g(x) could be negative?
First require g(x) > 0 for a strict less-than problem. Then square both sides, solve the new inequality, and intersect that result with both the domain and the positive-side condition.
How is solving a radical inequality different from a radical equation?
A radical equation usually checks isolated candidate roots. A radical inequality has intervals, sign branches, and endpoint rules, so verification must happen across ranges rather than only at one or two points.
Do you need to verify solutions after solving a radical inequality?
Yes. Verification is not optional because squaring is not always an equivalent move. The final answer should satisfy the original inequality, not only the squared version.