Build the inequality in the display at the top: click the Var, Num and Op buttons, one of the four Ineq symbols, a Rad button for a square, cube or fourth root, and the brackets, or type on the keyboard; wrap a multi-term radicand in brackets. Learn more about building inequalities
Clear empties the display, the curved arrow undoes the last edit and the ⌫ button deletes the character before the caret. Learn more about editing
Click Solve, or press Enter in the display, once the inequality is complete; an expression without an inequality symbol or without a radical is refused. Learn more about solving
The right panel lists the steps from isolating the radical and raising both sides to a power to the domain step, and the answer card gives the inequality and the interval; the box under the solver explains the method. Learn more about the solution
An even root compared with a negative number ends in No solution or in the whole domain, without any squaring. Learn more about these cases
√
Solve inequalities with square roots, cube roots, and fourth roots
Enter inequality...
Var
Num
Op
Ineq
Rad
Select an inequality type or enter your own, then click Solve to see the step-by-step solution.
How the solver works
The solver reads one inequality in x, y or n with the variable under a square, cube or fourth root on one side. It isolates the radical, settles the special cases of an even root compared with zero or a negative number, raises both sides to the index of the root, solves the linear inequality that remains, and, for an even root in a less-than form, intersects the result with the domain where the radicand is non-negative. The answer is shown as an inequality and as an interval. Learn more about radical inequalities
The solver shows an inequality display at the top with a blinking yellow caret marking the cursor position. Click anywhere in the display to place the cursor, or use arrow keys, Home, and End for keyboard navigation. Type directly or use the button panel below.
Three quick experiments:
• Type x>3 and press Enter — the right panel squares both sides to get x>9. Because the right side is positive both sides are non-negative, so the direction is preserved. • Click the "Linear Radicand" example template — an inequality with ax+b loads. The solver squares both sides, solves the linear inequality, and intersects with the domain if the inequality is less-than. • Try the "Cube Root" template to see how the solver handles 3x. No domain restriction and no direction-flip concerns because the cube function is strictly increasing.
The Solve button is disabled until you enter something. The dedicated "Rad" row inserts the three radical symbols: , 3, 4. An expression without an inequality symbol is refused with "Input must be an inequality (use <, >, ≤, or ≥)", one without a radical with "No radical found in the inequality", and radicals on both sides are not yet supported. x+5<2 isolates to x<−3 and ends in No solution. What makes an inequality radical is explained in what is a radical inequality.
Building Inequalities with Buttons and Keyboard
DemoA bracketed radicand, undo, delete, the domain step
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The button panel groups inputs by type. All inputs can be made either by clicking or typing on the keyboard.
• Variable row — x, y, n. • Number row — digits 0 through 9 and the decimal point. • Operator row — the caret operator for exponents, multiplication, division, plus, minus. • Ineq row — the four inequality symbols. • Rad row — the three radical symbols: square root, cube root, fourth root. • Bracket row — parentheses for grouping the radicand.
To enter 2x+1, click the square-root button then open a parenthesis, type the radicand, and close the parenthesis. The solver requires parentheses around any multi-term radicand to parse it correctly.
Keyboard shortcuts: type letters and digits directly, use star or slash for multiplication and division, type < and > directly, type <= or >= to insert the combined symbols ≤ and ≥, and press Enter to solve. The Unicode radical symbols can also be typed or pasted directly. The Clear button empties the display entirely; the curved-arrow button steps through the undo stack one edit at a time; the ⌫ button deletes the character before the caret. 2x+1≤3 squares to 2x+1≤9 and, after the domain 2x+1≥0 is applied, gives [−0.5,4]. Why the domain must be applied is explained in domain restrictions and the even-root trap. Ctrl+Z undoes up to fifty edits back.
Try an Example — Eight Form Templates
DemoOpen the templates, load two forms
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Click the "Try an Example" header to expand the template panel. Each card generates a random inequality of that form. Clicking again produces a new random version.
• Simple — x>c or x<c. The base case with a single-variable radicand. • Linear Radicand — ax+b>c. Tests the linear-equation solving inside the radical after squaring. • With Constant — x+c>d. Requires subtracting c to isolate the radical before squaring. • With Coefficient — a⋅x>c. Tests division by the coefficient to isolate the radical. • Cube Root — 3x>c. No domain restriction; cubing both sides directly produces the answer. • Compare to Zero — ax+b>0 or <0. Exercises the special-case branches for zero and negative right sides. • Cube Root Linear — 3ax+b>c. Cube both sides, solve the resulting linear inequality. • Fourth Root — 4x>c. Even root with index 4; same logic as square root but raise to the fourth power.
Roughly 80 percent of generated inequalities produce clean integer boundaries. Loading a template clears the previous solution; click Solve to work the new inequality. The four stages applied to every template are listed in the solving process explained.
Reading the Step-by-Step Solution
DemoA cube root, then a coefficient to divide out
Step 0 of 3
The solution panel shows each algebraic move as a labeled step. The exact sequence depends on the structural case.
• Rearrange Inequality (when applicable) — if the radical is on the right, the solver swaps sides and flips the direction. • Evaluate Constant — simplifies the non-radical side to a single number. • Isolate Radical Term — subtracts additive constants or divides by a coefficient outside the radical to get the radical alone on one side. • Radical Properties (for special cases) — handles the cases where the right side is zero or negative against an even-root left side. Determines all-reals, no-solution, single-point, or domain-restriction outcomes without ever raising to the power. • Square Both Sides or Cube Both Sides or Raise Both Sides — the central move. Removes the radical by raising both sides to its index. Direction is preserved because the operation is on non-negative quantities (even roots) or monotonically increasing (odd roots). • Solve Linear Inequality — handles the resulting polynomial inequality in the variable. • Apply Domain Restriction (for even-root less-than) — intersects the algebraic solution with the requirement that the radicand be non-negative.
The final-answer card shows the answer in two forms: the inequality form (such as a≤x<b) and the corresponding interval notation. 3x−1>2 is cubed to x−1>8 and gives x>9 with no domain step; 2⋅x≥4 is divided by 2 first and gives x≥4, the interval [4,∞). The box under the solver, How the solver works, summarises the method in a few lines and links to the theory below. When the domain step is needed and when it is not is explained in domain restrictions and the even-root trap.
Domain Restrictions and the Even-Root Trap
The single biggest source of error in radical inequality solving is forgetting the domain restriction for even roots.
The principle. An even root is defined only when the radicand is non-negative:
• f(x) is defined when f(x)≥0. • 4f(x) is defined when f(x)≥0. • Any value of x outside this set makes the left side undefined and cannot be part of the solution.
Where the trap hits. When solving f(x)<c with c>0, squaring gives f(x)<c2. But every solution must also satisfy f(x)≥0 to make the original radical meaningful. The complete solution is
0≤f(x)<c2
which is the intersection of the squared inequality and the domain. The solver builds this intersection automatically.
Where the trap does not hit. For f(x)>c with c≥0, squaring gives f(x)>c2. This already implies f(x)>0, so f(x)≥0 is automatic and no separate intersection is needed. The solver recognizes this and skips the domain intersection.
Odd roots are immune. Cube roots and other odd roots are defined for every real number. There is no domain restriction and no trap to worry about. The solver handles cube root and fourth root cases differently in this respect.
Special cases the solver handles automatically.
• f(x)<c with c<0: no solution. A non-negative quantity cannot be less than a negative number. • f(x)>c with c<0: solution is the entire domain. The radical is always non-negative, so the inequality holds whenever the radical is defined. • f(x)≥0: solution is the entire domain. • f(x)>0: solution is the domain minus the points where the radicand is zero.
For deeper coverage see domain restrictions and principal square root.
A radical inequality is an inequality that contains at least one radical expression where the variable appears under the radical sign. The general form for a single-radical inequality is
nf(x)□g(x)
where n≥2 is the index, f(x) is the radicand, g(x) is the other side (often a constant), and □ is one of the four inequality symbols.
The two regimes.
• Even-index radicals (n=2,4,6,…). The radical is defined only when the radicand is non-negative, and the radical itself is always non-negative. Solving involves both an algebraic step (raise both sides to the n-th power) and a domain step (intersect with the radicand-non-negative condition). • Odd-index radicals (n=3,5,7,…). The radical is defined for every real number and takes both positive and negative values matching the sign of the radicand. Solving requires only the algebraic step.
Solution shapes. Depending on the structural case, a radical inequality can produce:
• A single ray (most common with odd roots). • A bounded interval (common with even-root less-than). • A half-line starting at the domain boundary. • A single point (degenerate non-strict case). • All real numbers (even-root ≥0, or odd-root with always-true reduction). • No solution (even-root strictly less than a negative number).
The pivotal move. Raising both sides to the index n removes the radical. This move preserves direction when both sides are non-negative (even-root case after isolation) or when the function is increasing (odd-root case). Both of these conditions are sustained by the structural setup, so no direction-flip occurs in standard radical inequality solving.
For deeper coverage see radical inequality, principal nth root, and interval notation.
Four stages handle any standard single-radical inequality.
• Stage 1: Isolate the radical. Move the radical to one side. Subtract additive constants outside the radical. Divide by any coefficient multiplying the radical. The goal is to reach the form nf(x)□c where c is a number.
• Stage 2: Examine the special cases for even roots.
• c<0 with less-than form: no solution. • c<0 with greater-than form: solution is the entire domain (radicand ≥0). • c=0 with strict less-than: no solution. • c=0 with non-strict less-than: solution is the single set where radicand equals zero. • c=0 with strict greater-than: solution is the domain minus zero-radicand points. • c=0 with non-strict greater-than: solution is the entire domain.
If none of these apply, proceed to stage 3.
• Stage 3: Raise both sides to the n-th power. The inequality becomes f(x)□cn. Direction is preserved.
• Stage 4: Solve and intersect with the domain.
• Odd roots: solve f(x)□cn directly. The result is the final answer. • Even roots, □ is greater-than form: solve f(x)□cn. Domain f(x)≥0 is automatically satisfied because cn≥0. • Even roots, □ is less-than form: solve f(x)□cn, then intersect with f(x)≥0. The result is a bounded interval rather than a ray.
The implementation. Internally, the solver navigates this decision tree based on the index, the inequality direction, and the sign of c. The output ranges from a simple ray to a compound bounded interval depending on which branch applies.
For comprehensive treatment see solving radical inequalities and squaring both sides.
Radical Equation — the equality counterpart nf(x)=g(x). Solving introduces a check-for-extraneous-solutions step because squaring both sides can introduce false roots. For inequalities the domain intersection plays a similar role.
Principal Square Root — the convention that a denotes the non-negative root. Without this convention, 4 could mean either 2 or −2; the convention fixes it as 2, making the radical expression a well-defined function and enabling the sign analysis used in inequality solving.
Domain of a Function — the set of inputs for which the function is defined. For even-root expressions, the domain restriction shapes the final solution set.
Cube Root — the inverse of cubing. Defined for every real number, takes negative values for negative inputs. The most prominent odd-index radical and the easiest case to handle.
Nth Root — the generalization to arbitrary integer index. For odd n, behaves like the cube root. For even n, behaves like the square root.
Quadratic Inequality — the polynomial counterpart that often arises when squaring a radical with a linear radicand and combining. The solver routes the post-squaring step through the same linear or polynomial-inequality logic.
Compound Inequality — a chain like a≤x<b that captures a bounded interval. The intersection of the algebraic solution and the domain restriction in even-root less-than cases naturally produces a compound inequality.
Extraneous Solution — a value satisfying the squared equation but not the original radical equation. The inequality analogue is a value satisfying the post-squaring inequality but lying outside the domain; the domain intersection filters these out.
Squaring Both Sides — the core algebraic move. Preserves inequality direction when both sides are non-negative; can break it otherwise. The solver enforces non-negativity through the isolation step before squaring.
Interval Notation — the standard way to express the solution. Parentheses for excluded endpoints, brackets for included endpoints, infinity always with parentheses.
Rational Exponent — the alternative notation f(x)1/n for nf(x). The inequality f(x)1/n≥c is identical to nf(x)≥c; the underlying analysis is the same.