Build the inequality in the display at the top: click the Var, Num and Op buttons, one of the four Ineq symbols, the x² or ^ power buttons and the brackets, or type on the keyboard; the ÷ button writes a fraction, so wrap a multi-term denominator 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 variable in a denominator, is refused. Learn more about solving
The right panel lists the steps from the excluded values and the combined fraction to the critical points and the sign chart, and the answer card gives the intervals; the box under the solver explains the method. Learn more about the solution
Denominator zeros are never included in the answer; numerator zeros are included only for ≤ and ≥. Learn more about the endpoints
¹/ₓ
Solve rational inequalities with sign charts, step by step
Enter inequality...
Var
Num
Op
Ineq
Select an inequality type or enter your own rational inequality, 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 in at least one denominator. It lists the excluded values, moves every term to the left and combines them into a single fraction over the common denominator, finds the zeros of the numerator and of the denominator, tests the sign of the fraction on each interval between them, and keeps the intervals whose sign matches. Numerator zeros are included for leq and geq; denominator zeros never are. Learn more about rational 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 1/x>2 and press Enter — the right panel combines to (1−2x)/x>0, identifies critical points at x=0 (excluded, denominator) and x=1/2 (numerator zero), builds the sign chart, and reports the solution as (0,1/2). • Click the "Fraction less than or equal to 0" example template — an inequality of the form (x+a)/(x+b)≤0 loads. Note how the solver uses brackets for the numerator-zero endpoint and parentheses for the denominator-zero endpoint. • Try the "Sum of Fractions" template to see how the solver finds a common denominator before identifying critical points.
The Solve button is disabled until you enter something. The dedicated "Ineq" row inserts the four inequality symbols. An expression without one of them is refused with "Input must be an inequality (use <, >, ≤, or ≥)", and one without a variable in a denominator with "This is not a rational inequality". 1/x>0 holds exactly for x>0, the interval (0,∞). What makes an inequality rational is explained in what is a rational inequality.
Building Inequalities with Buttons and Keyboard
DemoBracketed numerator and denominator, undo, delete, Solve
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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 — multiplication, division (the division operator builds fractions in the display), plus, minus. • Ineq row — the four inequality symbols. • Special row — the x2 shortcut button, caret operator for arbitrary exponents, and parentheses for grouping.
The division operator is the key tool for this solver: typing 1/x creates a fraction structure that the solver recognizes as a rational expression. For multi-term denominators, use parentheses: 1/(x+2) is parsed as one over the quantity x+2, not as 1/x+2.
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. Ctrl+Z undoes up to fifty edits back. 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. (x+1)/(x−2)≤0 is typed with both brackets and gives [−1,2): the numerator zero is included, the denominator zero is not. The endpoint rule is explained in the sign-chart method.
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 Reciprocal — 1/x>a. The most basic rational inequality. Critical points are 0 (denominator) and 1/a (numerator after combining). • Linear Denominator — a/(x+b)<c. Standard one-fraction inequality with a linear denominator. • Fraction ≤0 — (x+a)/(x+b)≤0. Single fraction compared to zero, no combining needed. • Quadratic Numerator — (x2+bx+c)/(x+d)>0. Three critical points (two from the numerator quadratic, one from the denominator); four intervals in the sign chart. • Sum of Fractions — 1/x+1/(x+a)<b. Requires combining two fractions over a common denominator before the sign chart. • Proportion Inequality — a/(x+b)>c/(x+d). Compare two fractions; subtraction combines them. • Always Positive — a/x2>0. The denominator x2 is non-negative, so the only excluded value is x=0 and the answer is all reals except 0. • Complex Rational — (ax+b)/((x+c)(x+d))≥0. Three critical points and shows how multiple denominator factors interact.
Roughly 80 percent of generated inequalities produce clean integer critical points. Loading a template clears the previous solution; click Solve to work the new inequality. The five stages applied to every template are listed in the solving process explained.
Reading the Step-by-Step Solution
DemoThree critical points, then an always-true case
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The solution panel shows each algebraic move as a labeled step.
• Original Inequality — restates the input. • Excluded Values — lists every value of the variable that makes any denominator zero. These are forbidden in both the problem and the answer. • Combine into Single Fraction — the solver moves everything to one side and combines into a single rational expression over the least common denominator. • Combined Form — displays the combined inequality P(x)/Q(x)□0 where P is the combined numerator and Q is the LCD. • Find Critical Points — lists numerator zeros and denominator zeros separately. • Sign Chart — shows the sign of P(x)/Q(x) on each interval determined by the critical points. The chart displays one + or − per interval. • Solution — gives the final answer in interval notation, the union of intervals whose sign matches the inequality direction.
Numerator zeros under a non-strict inequality use brackets (included). All denominator zeros use parentheses (excluded) regardless of inequality strictness. The final-answer card shows the interval notation directly. 1/x+1/(x+2)<0 combines to (2x+2)/(x(x+2))<0 with the critical points −2, −1 and 0 and the answer (−∞,−2)∪(−1,0); 1/x2>0 is Always True and the answer is all real numbers except the excluded 0. The box under the solver, How the solver works, summarises the method in a few lines and links to the theory below. Why the sign can change only at critical points is explained in the sign-chart method.
The Sign-Chart Method — Why It Works
A rational expression P(x)/Q(x) can change sign only at points where the numerator or denominator equals zero. Between consecutive such points the sign is constant. The sign chart records the sign on each interval.
Why sign changes only happen at critical points.
• At a numerator zero, the rational expression equals zero. As x crosses such a point, the numerator changes sign (assuming the root has odd multiplicity), so the expression flips sign. • At a denominator zero, the rational expression is undefined — it has a vertical asymptote or hole. As x crosses such a point, the denominator changes sign (assuming odd multiplicity), and again the expression flips sign. • Between critical points, both numerator and denominator have constant sign, so the quotient does too.
Testing the sign on each interval. Plug any value from the interval into the expression. The sign of the result is the sign on that interval. The solver picks a midpoint or a value outside the outer critical points.
Endpoint rules.
• Numerator zero with non-strict inequality — included (use bracket). The expression equals zero, which satisfies ≤0 or ≥0. • Numerator zero with strict inequality — excluded (use parenthesis). Zero does not satisfy strict <0 or >0. • Denominator zero, any inequality — excluded (use parenthesis). The expression is undefined.
Repeated factors and bouncing. If a factor like (x−2)2 appears, the expression does not change sign as x crosses 2 because the squared factor has even multiplicity. The solver detects this through the test-point evaluation, so the resulting sign chart is correct without special-case logic.
For deeper coverage see sign chart and rational function.
A rational inequality is a comparison involving a rational expression (a quotient of two polynomials) and one of the four inequality symbols. The general form is
Q(x)P(x)□c
where P and Q are polynomials, Q contains the variable, and □ is one of <, >, ≤, ≥. By moving c to the left and combining, every rational inequality reduces to the form R(x)□0 where R is a single rational expression.
Domain. The domain of a rational expression excludes every value of the variable that makes any denominator zero. These excluded values can never be part of the solution and must be carefully tracked.
Solution shape. The solution to a rational inequality is a union of intervals on the real line. The number of intervals depends on the number of critical points (numerator zeros plus denominator zeros). With k critical points there can be at most k+1 intervals to choose from.
Why rational inequalities are harder than rational equations. Solving the equation P/Q=0 amounts to finding the numerator zeros (excluding any that are also denominator zeros). Solving the inequality requires the full sign analysis: it is not enough to know where the expression equals zero; you must know where it is positive, negative, and undefined.
Why rational inequalities are harder than polynomial inequalities. Polynomial inequalities have only zeros as critical points and are defined everywhere. Rational inequalities add a second class of critical points (denominator zeros) and forbidden values where the expression has no value at all.
For deeper coverage see rational inequality, rational expression, and excluded values.
• Stage 1: Move everything to one side. Subtract or add to make the right side zero. The inequality becomes R(x)□0.
• Stage 2: Combine into a single fraction. Find a common denominator for all the rational terms on the left side and add them together. The result is a single rational expression P(x)/Q(x). Do not clear the denominator: that step is invalid in inequalities because it requires knowing the sign of the denominator.
• Stage 3: Find the critical points.
• Numerator zeros — solve P(x)=0. • Denominator zeros — solve Q(x)=0. These are excluded values; they cannot be solutions but they still divide the number line.
• Stage 4: Build the sign chart. Order the critical points on the number line. Test one value in each resulting interval to determine the sign of P(x)/Q(x) there.
• Stage 5: Select intervals matching the inequality.
• R(x)>0: choose intervals with sign +. • R(x)<0: choose intervals with sign −. • R(x)≥0: choose + intervals plus numerator zeros (bracket); exclude all denominator zeros (parenthesis). • R(x)≤0: choose − intervals plus numerator zeros (bracket); exclude all denominator zeros (parenthesis).
The implementation. Internally, the solver represents polynomials as coefficient arrays, finds roots via factoring and numerical methods, and constructs the sign chart by evaluating the combined expression at midpoints. The final-answer card formats the result as a union of intervals with correct bracket-or-parenthesis logic per endpoint.
For comprehensive treatment see solving rational inequalities and sign analysis.
Rational Expression — the quotient of two polynomials. The object that lives on one side of a rational inequality.
Rational Function — the function f(x)=P(x)/Q(x) whose graph contains vertical asymptotes at denominator zeros and crosses the x-axis at numerator zeros (where they do not coincide with denominator zeros).
Sign Chart — the tabular method for tracking signs across critical points. Foundational for rational, polynomial, and any other inequality involving a sign question on intervals.
Excluded Values — values of the variable that make any denominator zero. They are forbidden from the domain and therefore forbidden from any solution set.
Critical Points — values where the sign of the expression can change. For rational inequalities these are the numerator zeros and the denominator zeros.
Vertical Asymptote — the geometric counterpart of a denominator zero in the graph of a rational function. The expression goes to plus or minus infinity as the variable approaches the asymptote.
Polynomial Inequality — the special case with constant (nonzero) denominator. The same sign-chart logic applies, with only numerator zeros as critical points.
Quadratic Inequality — the second-degree polynomial case. Often appears as the numerator or factor of a rational inequality.
Linear Inequality — the first-degree polynomial case. A simple rational inequality after combining might reduce to a linear inequality structure.
Interval Notation — the way the answer is written. Parentheses exclude endpoints, brackets include them, the union symbol ∪ joins disjoint pieces.
Common Denominator — the technique for combining multiple fractions. Same idea as adding rational numbers, but with polynomials in place of integers.
Factoring — the algebraic tool for finding polynomial roots cleanly. Fully factored numerator and denominator make the critical points immediate by inspection.