Solve any rational inequality instantly using the sign chart method — get step-by-step solutions with critical points identified, animated sign charts, number line graphs, function graphs, and interval notation. Free, no sign-up required.
Why this page is different
Rational Solver
Built for rational inequalities with standard-form rewrites, critical points, sign charts, excluded denominator values, and graph-based verification.
Split numerator and denominator input
Build the fraction first, then compare it with any value.
Preview
((x+1))/((x-2)) > 0
Supported Input Styles
(x+1)/(x-2)>0 solves a standard-form rational inequality.(x-3)/(x+4)<=0 keeps a numerator zero but excludes the denominator root.(x^2-4)/(x+3)>0 factors the numerator into linear pieces.(2x-1)/((x+1)(x-3))>=0 handles multiple undefined points.1/(x-5)>2 is moved to one side automatically.Math Keyboard
Tap fraction symbols, powers, and comparison signs for fast rational-inequality input.
Result
The sign chart marks numerator zeros and denominator restrictions, then keeps only the intervals whose sign matches the inequality.
Standard form
Numerator zeros
-1
Undefined points
2
Solution
Factor view
Numerator
Denominator
Step 1
Identify the rational structure
A rational inequality is solved by tracking the sign of the numerator and denominator across the real line.
Before
After
Step 2
Check the standard form
The right side is already 0, so the inequality is ready for sign-chart analysis.
Before
After
Step 3
Factor when useful
Factoring exposes the real zeros that control the sign chart and makes repeated roots easier to recognize.
Before
After
Step 4
Find critical points
Zeros of the numerator can become included endpoints for ≤ or ≥. Zeros of the denominator are undefined points and are never included.
Before
After
Step 5
Build the sign chart
Each critical point splits the number line into intervals. One test value per interval is enough because the sign can only change at a zero or undefined point.
Before
After
Step 6
Apply endpoint rules
The inequality is strict, so numerator zeros stay open. Denominator zeros always stay excluded.
Before
After
Step 7
Write the solution
Keep only the intervals whose sign matches the inequality, then convert the result into interval notation.
Before
After
Sign Chart
Each interval keeps a constant sign, so one test value per interval is enough.
| Row | (-∞, -1) | (-1, 2) | (2, +∞) |
|---|---|---|---|
| Test value | |||
x + 1 numerator | - | + | + |
x - 2 denominator | - | - | + |
Overall sign f(x)/g(x) | + | - | + |
Solution Keep matching intervals |
Calculator Types
Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.
Solve quadratic inequalities with sign analysis, roots, interval notation, and a number-line graph.
Solve absolute value inequalities with case splitting, interval notation, and step-by-step explanations.
Solve compound inequalities with interval intersection, union logic, and graph output.
Focused on sign charts, undefined points, function graphs, and interval notation.
Graph linear, quadratic, and absolute value inequalities in two variables with shading, dashed or solid boundaries, and system overlap tools.
Zone 4
This rational inequality calculator is designed for the exact trouble spots students hit in algebra and precalculus: quotient inequalities such as (x + 1)/(x - 2) > 0, non-strict comparisons such as (x - 3)/(x + 4) <= 0, factorable quadratic numerators like (x^2 - 4)/(x + 3) > 0, multi-critical-point cases such as (2x - 1)/((x + 1)(x - 3)) >= 0, and rearrangement problems such as 1/(x - 5) > 2. You can type the whole inequality directly or switch to the split numerator and denominator builder when you want to see the fraction structure more clearly.
Once the expression is recognized, the page keeps the solving workflow close to what a strong teacher would do on paper. First put everything on one side so the comparison is against 0. Next identify zeros of the numerator and undefined points from the denominator. Then use the sign chart to test one value per interval. After that, move through the number line, function graph, and interval notation tabs so the answer is visible in several different representations instead of staying trapped in symbols.
The result area is built to solve two different jobs. If you are learning, follow the tabs in order: Steps, Sign Chart, Number Line, Function Graph, Interval Notation, Verify. If you are only checking homework, jump straight to the Sign Chart or Verify tab and test one value from inside the answer, one value outside it, and one undefined point. That three-value check catches almost every endpoint mistake.
Enter a rational inequality such as (x + 1)/(x - 2) > 0, switch to fraction-builder mode, or load one of the example buttons.
Review the standard-form rewrite, numerator zeros, denominator restrictions, sign chart, number line, and function graph together.
Use the interval notation and verification tabs to confirm which endpoints stay open, which numerator zeros are included, and which denominator points are always excluded.
A rational inequality compares a rational expression, meaning a quotient of polynomials, with zero or with another expression. In its standard classroom form it looks like f(x)/g(x) > 0, f(x)/g(x) < 0, f(x)/g(x) >= 0, or f(x)/g(x) <= 0. The core difference from linear or quadratic inequalities is that the denominator matters twice: it affects the sign, and it also creates values where the expression is undefined.
That undefined-domain issue is what makes rational inequalities conceptually different from ordinary polynomial sign problems. A numerator zero is a place where the rational expression equals 0. A denominator zero is a place where the expression does not exist at all. Those two kinds of critical points are not interchangeable, and the final number-line picture has to keep them visually separate.
The final answer to a rational inequality is usually an interval or a union of intervals. It can also be a single point, no solution, all real numbers, or all real numbers except one or more undefined points. That range of answer shapes is exactly why sign charts, number lines, and interval notation belong on the same page.
Model form
Key restriction
The most important idea in rational inequality work is that not all critical points behave the same way. A zero of the numerator makes the whole fraction equal to 0, so it may become an included endpoint if the inequality uses <= or >=. A zero of the denominator makes the fraction undefined, so it can never be included, no matter what comparison symbol appears in the problem.
Both kinds of points split the number line into sign intervals. That is why they both appear on the sign chart. But they get different endpoint rules. Students often remember to find both sets of points but then treat them identically in the final answer. That is the error this page is designed to prevent.
There is one more subtlety: repeated factors. If the numerator has a factor such as (x - 1)^2, the point x = 1 is still critical, but the sign does not flip there because an even power stays nonnegative on both sides. The same even-power idea can happen in the denominator too. A repeated denominator root is still undefined, still excluded, and still may fail to flip the sign.
| Comparison point | Zero of numerator | Zero of denominator |
|---|---|---|
| Function value | Equals 0 | Undefined because the denominator is 0 |
| Strict inequality | Excluded with an open endpoint | Always excluded |
| Inclusive inequality | Included if it satisfies the sign | Still excluded |
| Number-line marker | Open or closed circle | Vertical warning marker |
| Sign behavior | May flip for odd multiplicity | May flip for odd multiplicity, but never enters the solution |
The sign chart method is the standard reliable way to solve rational inequalities. It works because the sign of a rational expression can only change at critical points: zeros of the numerator and zeros of the denominator. Between those points, the sign stays constant, so testing one number per interval is enough.
This approach is much safer than cross-multiplying blindly. If you multiply both sides by a denominator whose sign is not fixed, you risk flipping the inequality in some intervals but not others. The sign chart method avoids that trap by keeping the denominator in view until the end.
Even when the numerator or denominator does not factor perfectly, the same workflow applies. You still move everything to one side, identify the real critical points, divide the number line into intervals, and read the sign on each interval. When the denominator is always positive or always negative, the chart becomes simpler, but the endpoint rules stay the same.
Sign multiplication rule
Working model
Zeros and undefined points split the real line into intervals. The sign chart chooses the positive or negative regions that match the inequality.
Rewrite the inequality so the right side is 0. If the problem starts as 1/(x-5) > 2, move the 2 left first and combine the result over a common denominator.
Factor the numerator and denominator when real factors are available. Factoring makes zeros, repeated roots, and sign changes much easier to see.
Set numerator factors equal to 0 to get zeros. Set denominator factors equal to 0 to get undefined points. Sort every critical point from left to right.
If there are n critical points, they create n+1 intervals. Each interval keeps a constant sign until another critical point is crossed.
Substitute one test value into each interval. Record the sign of each factor, then combine the signs to get the sign of the full rational expression.
Keep the intervals whose sign matches the inequality. Include numerator zeros only for <= or >=. Exclude denominator zeros every time.
Sign Chart
Each interval keeps a constant sign, so one test value per interval is enough.
| Row | (-∞, -1) | (-1, 2) | (2, +∞) |
|---|---|---|---|
| Test value | |||
x + 1 numerator | - | + | + |
x - 2 denominator | - | - | + |
Overall sign f(x)/g(x) | + | - | + |
Solution Keep matching intervals |
This is the classic two-critical-point introduction. One numerator zero and one denominator restriction split the real line into three intervals.
Critical points
x = -1 (zero), x = 2 (undefined)
Answer
Strict inequality, one zero, one undefined point.
This example shows the most common endpoint mistake: x = 3 can be included, but x = -4 cannot, even though the nearby interval is valid.
Critical points
x = -4 (undefined), x = 3 (zero (included))
Answer
≤ includes a numerator zero but still excludes the denominator root.
Factoring the numerator into two linear factors creates three critical points and four sign intervals.
Critical points
x = -3 (undefined), x = -2 (zero), x = 2 (zero)
Answer
Quadratic numerator and multiple sign flips.
A linear numerator with two denominator factors creates several intervals and tests whether the inclusive symbol adds the numerator zero without ever admitting denominator roots.
Critical points
x = -1 (undefined), x = 0.5 (zero (included)), x = 3 (undefined)
Answer
Multiple undefined points and a non-strict inequality.
This is the model problem for showing why standard form matters. You cannot read a sign chart correctly until every term has been moved to one side.
Critical points
x = 5 (undefined), x = 5.5 (zero)
Answer
Rearrangement into quotient compared with 0.
Since x^2 + 1 is always positive, the sign chart collapses to the sign of the numerator alone.
Critical points
x = -2 (zero)
Answer
Constant-sign denominator.
The repeated numerator root creates a critical point that does not flip the sign. The denominator restriction still does.
Critical points
x = -2 (undefined), x = 1 (zero)
Answer
Even-powered factor and non-flipping sign.
Both polynomials factor, and one critical point appears in both the numerator and denominator. That overlap is a good reminder that undefined still beats zero.
Critical points
x = -2 (zero and undefined), x = 2 (undefined), x = 3 (zero (included))
Answer
Shared critical point and mixed endpoint logic.
A rational inequality number line must show more information than a linear one. Numerator zeros may get open or closed circles depending on whether the inequality is strict or inclusive. Denominator zeros are different: they should be marked as excluded warning points because the expression is undefined there.
This distinction is not cosmetic. It is the visual version of the domain rule. If two intervals are both valid but a denominator root sits between them, the graph must break there. That is why interval notation for rational inequalities often uses a union, even when the expression seems positive on both sides of one missing point.
Number Line
Zero points can open or close. Undefined denominator points stay excluded and use an orange marker.
Interval notation for rational inequalities follows two different endpoint rules at once. Numerator zeros can be included only when the symbol is <= or >= and the rational expression is actually defined there. Denominator zeros are never included because division by zero is undefined.
A strong final check is to read the answer aloud with the graph in mind. For example, (-infinity, -4) union (-4, 3] means every x less than -4 or between -4 and 3, with -4 excluded and 3 included. If the spoken sentence does not match the number line, the interval notation needs one more correction.
| Solution shape | Example | Interval notation | Meaning |
|---|---|---|---|
| Two open outer intervals | Strict sign with both endpoints excluded. | ||
| Mixed union around an undefined point | The numerator zero can close, but the denominator zero stays open. | ||
| Single bounded interval | Strict inequality between two critical points. | ||
| All reals except one undefined point | Every interval works, but the denominator root still breaks the line. | ||
| No solution | No interval matches the requested sign. |
Expressions such as (x + 2)/(x^2 + 1) < 0 become easier because the denominator never changes sign. The solution comes entirely from the numerator, but the method is still a rational sign chart method because the denominator analysis is what tells you the sign is fixed.
If every interval on the sign chart has the wrong sign, the answer is the empty set. A common example is (x^2 + 1)/(x^2 + 4) < 0, where both numerator and denominator stay positive for all real x.
Sometimes every interval satisfies the inequality, but denominator roots still have to be removed from the domain. In that case the final answer is all real numbers with one or more exclusions.
Factors such as (x - 1)^2 create critical points without forcing a sign change. They still matter for endpoint logic, but they do not automatically flip the sign chart the way a simple first power does.
A rational inequality compares a rational expression, such as (x + 1)/(x - 2), with another value using <, >, <=, or >=. Its solution is usually a set of intervals rather than one isolated number.
A linear inequality has no variable in the denominator, so its main job is to isolate x. A rational inequality also has to respect domain restrictions because denominator values equal to zero are undefined.
Critical points are the real numbers where the numerator is zero or the denominator is zero. They divide the real line into intervals where the sign of the rational expression stays constant.
Zeros come from the numerator and may be included when the inequality is inclusive. Undefined points come from the denominator and are never included because division by zero is undefined.
Because a rational expression does not exist when its denominator is zero. Any x-value that makes the denominator 0 must be excluded from the domain and from the final solution.
Include an endpoint only when it comes from a numerator zero and the inequality uses <= or >=. Denominator zeros are excluded in every case.
The comparison symbol does not change the domain. If the denominator is zero, the rational expression is undefined, so the point cannot be part of the solution set.
That point becomes a break in the number line. The solution may continue on both sides, but the point itself is excluded and shown separately from ordinary endpoints.
An even-powered factor such as (x - 1)^2 does not flip sign when you cross its root. The point is still critical, but the sign on the two adjacent intervals may stay the same.
The sign chart method finds numerator zeros and denominator restrictions, splits the number line into intervals, tests one value per interval, and keeps the intervals whose sign matches the inequality.
First move everything to one side so the inequality compares a quotient with 0. Then factor, list the critical points, choose one test value from each interval, record the sign of each factor, and combine the signs to read the result.
Choose any convenient number strictly inside each interval. Midpoints are common because they avoid the endpoints and usually keep the arithmetic simple.
After you know the sign on every interval, keep the intervals with the required sign. Then apply the endpoint rules: inclusive numerator zeros may enter, denominator zeros never do.
Factoring is very helpful because it makes the critical points explicit, but the real goal is sign information. If a factor does not split nicely, you can still analyze its sign from roots or constant-sign behavior.
Put the inequality into quotient-versus-zero form, factor when possible, find numerator zeros and denominator restrictions, build the sign chart, choose the satisfying intervals, and then apply endpoint rules.
Move every term to one side first. For example, 1/(x - 5) > 2 becomes (11 - 2x)/(x - 5) > 0 before you build the sign chart.
Treat them the same way after factoring or analyzing their real roots. Quadratic factors create additional critical points, and repeated roots may affect whether the sign flips.
It has no solution when none of the sign-chart intervals satisfy the requested sign. That often happens when both numerator and denominator keep the same sign everywhere.
It happens when every interval satisfies the inequality and there are no denominator restrictions. If denominator roots exist, the result becomes all real numbers except those undefined points.
The denominator no longer creates sign flips, so the sign of the rational expression is determined entirely by the numerator.
Enter a rational inequality, press Solve, then move through the Steps, Sign Chart, Number Line, Function Graph, Interval Notation, and Verify tabs. Example buttons are included if you want a model problem first.
Yes. The full solver, sign chart, graph views, interval notation, and verification tools are free to use without registration.
Keep Exploring
Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.
Solve absolute value inequalities with case splitting, interval notation, and step-by-step explanations.
Solve compound inequalities with interval intersection, union logic, and graph output.
Solve quadratic inequalities with sign analysis, roots, interval notation, and a number-line graph.