Inequality Calculator
Solve linear, compound, absolute value, and quadratic inequalities instantly — with interval notation and a number line graph for every solution
Solve Any Inequality, Step by Step
Our inequality calculator handles linear, compound, absolute value, and quadratic inequalities — returning the solution set, interval notation, and a number line graph, with full step-by-step logic for every problem type.
Your Inequality Solution
More Than a Single Answer
Why students switch from manual sign-flip worksheets to our AI math solver engine.
| Capability | Manual Calculation | Inequality Calculator |
|---|---|---|
| Linear, compound, absolute value & quadratic modes | ✘ One method only | ✔ All 4 modes |
| Converts to interval notation | ✘ Manual conversion | ✔ Automatic |
| Flips sign on negative multiply/divide | ⚠ Easy to forget | ✔ Always exact |
| Number line graph of solution set | ✘ Drawn by hand | ✔ Included |
| Step-by-step explanation | Textbook only | ✔ Every problem |
What This Inequality Calculator Solves
Six core inequality types covered with full working, from middle-school basics to college-level applications.
Linear Inequalities
Isolate the variable and find every value that makes a one-variable inequality true, with each algebra step shown.
Example: 3x + 5 > 17 → x > 4
Compound Inequalities
Solve “and”/”or” combinations of two inequalities at once, returning a single combined solution range.
Example: -3 < 2x + 1 < 9 → -2 < x < 4
Absolute Value Inequalities
Split an absolute value inequality into its two underlying cases and solve each branch automatically.
Example: |2x − 4| ≤ 10 → -3 ≤ x ≤ 7
Quadratic Inequalities
Find the roots, test each interval, and determine where a quadratic expression is positive or negative.
Example: x² − 5x + 6 > 0 → x < 2 or x > 3
Interval & Set Notation
Every solution converts automatically between inequality notation, interval notation, and set-builder notation.
Example: x ≥ 4 → [4, ∞)
Rational & Polynomial Inequalities
Identify critical points, test the sign of each interval, and combine them into a full solution set — bridging into algebra and beyond.
Example: (x − 1)/(x + 2) > 0 → x < -2 or x > 1
Built for Every Inequality Problem, Every Level
Middle & High School
Learn how to isolate a variable and flip the sign correctly, with visual, step-by-step solutions that match how inequalities are taught in Algebra 1 and Algebra 2.
College Students
Verify compound, absolute value, and quadratic inequality answers fast, then move into rational inequalities and sign charts for precalculus.
Teachers & Tutors
Generate quick, accurate worked examples for lesson plans, homework checks, or explaining a sign flip on the fly.
How the Inequality Calculator Works
Get a complete inequality solution in four simple steps.
Choose Your Type
Pick linear, compound, absolute value, or quadratic — whatever your problem gives you.
AI Applies the Rules
The calculator isolates the variable, flipping the sign whenever it multiplies or divides by a negative number, showing each transformation.
Review Every Form
See the solution set, plus interval notation, set-builder notation, and a number line graph — all derived from the same answer.
Unlock the Full Breakdown
Get the complete step-by-step explanation and graph to confirm your homework or exam prep answer.
Frequently Asked Questions
Common questions about solving and graphing inequalities.
How do you solve a basic linear inequality?
Treat it like a linear equation: add, subtract, multiply, or divide both sides to isolate the variable. The only difference is that multiplying or dividing both sides by a negative number flips the inequality sign. Enter your inequality above and the calculator shows every step.
What does it mean to flip the inequality sign?
When you multiply or divide both sides of an inequality by a negative number, the direction of the comparison reverses: a less-than becomes greater-than, and vice versa. Forgetting this step is the most common mistake when solving inequalities by hand.
How do you solve a compound inequality?
For an “and” compound inequality like -3 < 2x + 1 < 9, perform the same operation across all three parts at once until the variable is isolated in the middle. For an “or” compound inequality, solve each piece separately and combine the resulting ranges.
How do you solve an absolute value inequality?
Split the inequality into two separate cases — one where the expression inside the absolute value is positive, and one where it’s negative — then solve each case and combine the results. The calculator’s Absolute Value tab handles both branches automatically.
How do you solve a quadratic inequality?
Find the roots of the related quadratic equation, use them to divide the number line into intervals, then test a value from each interval to see where the original inequality holds true. The calculator shows each interval test in the full breakdown.
What is interval notation and how is it different from inequality notation?
Inequality notation, like x > 4, describes a solution using comparison symbols. Interval notation, like (4, ∞), describes the same range using brackets and parentheses. Square brackets include the endpoint; parentheses exclude it. The calculator converts automatically between both.
Can this calculator graph the solution on a number line?
Yes. Once a solution set is found, the calculator shows it shaded on a number line, with an open circle for strict inequalities (< or >) and a closed circle for inclusive ones (≤ or ≥).
Does this work for rational and polynomial inequalities too?
Yes. The calculator identifies critical points where the numerator or denominator equals zero, tests the sign of each resulting interval, and combines the valid intervals into a final solution set.
How to Use the Inequality Solver for Homework & Exam Prep
An inequality calculator finds every value of a variable that makes a mathematical comparison true. Unlike an equation, which usually has one exact answer, an inequality describes a whole range of valid values — written with symbols like <, >, ≤, and ≥. This page covers every common type of inequality problem: linear, compound, absolute value, and quadratic, plus a look at rational and polynomial inequalities. It’s one of several free math solver tools built for exactly this kind of step-by-step practice.
Instead of returning only a final answer, this calculator shows the complete working: the rule applied at each step, where the sign flips, and the resulting solution set in three forms — inequality notation, interval notation, and a number line graph. That makes it useful both for checking a homework answer and for understanding the reasoning behind it.
How to Solve a Linear Inequality
A linear inequality looks almost identical to a linear equation, and the solving process is nearly the same. You isolate the variable using addition, subtraction, multiplication, and division. The one rule that makes inequalities different:
Multiplying or dividing both sides by a negative number flips the inequality sign.
For example, to solve 3x + 5 > 17: subtract 5 from both sides to get 3x > 12, then divide both sides by 3 (a positive number, so the sign stays the same) to get x > 4. Every value greater than 4 satisfies the original inequality.
Why the Sign Flips
Consider 2 > 1. Multiply both sides by -1 without flipping the sign and you’d get -2 > -1, which is false. Flipping the sign to -2 < -1 keeps the statement true. This same logic applies any time you multiply or divide both sides of an inequality by a negative value.
Solving Compound Inequalities
A compound inequality combines two inequality statements into one, joined by “and” or “or.”
“And” Compound Inequalities
Written as a single chained statement like -3 < 2x + 1 < 9, an “and” compound inequality requires both conditions to be true at once. Solve it by performing the same operation across all three parts simultaneously: subtract 1 from each part to get -4 < 2x < 8, then divide each part by 2 to get -2 < x < 4.
“Or” Compound Inequalities
An “or” compound inequality, like x < -1 or x > 5, is satisfied if either condition holds. These are solved by treating each inequality separately, then combining the two solution sets without requiring overlap.
Solving Absolute Value Inequalities
Absolute value measures distance from zero, so an absolute value inequality like |2x − 4| ≤ 10 must be split into two cases before it can be solved algebraically:
- Case 1 (positive):
2x − 4 ≤ 10 - Case 2 (negative):
-(2x − 4) ≤ 10, which becomes2x − 4 ≥ -10
Solving both cases and combining the results gives -3 ≤ x ≤ 7. For a “greater than” absolute value inequality, the combined solution is typically two separate ranges rather than one continuous range — the calculator shows which pattern applies based on the comparison symbol.
< or ≤, the solution is usually a single connected interval (“and” logic). If it uses > or ≥, the solution usually splits into two separate rays (“or” logic). Checking the symbol first tells you which structure to expect.
Solving Quadratic Inequalities
A quadratic inequality, such as x² − 5x + 6 > 0, asks where a parabola lies above or below the x-axis rather than for a single root. The standard method:
- Find the roots of the related equation
x² − 5x + 6 = 0, which arex = 2andx = 3. - Use the roots to split the number line into three intervals:
x < 2,2 < x < 3, andx > 3. - Test one value from each interval in the original inequality to see if it holds true.
- Combine the intervals that satisfy the inequality into the final solution:
x < 2 or x > 3.
Rational and Polynomial Inequalities
Rational inequalities — those involving a fraction with a variable, like (x − 1)/(x + 2) > 0 — and higher-degree polynomial inequalities follow the same interval-testing method used for quadratics, with one extra caution: values that make a denominator equal to zero must always be excluded from the solution set, since division by zero is undefined.
This bridges naturally into broader algebra work, where sign charts and critical-point analysis show up again in calculus when studying where a function is increasing, decreasing, positive, or negative.
Common Inequality Mistakes to Avoid
- Forgetting to flip the sign: always reverse the inequality symbol when multiplying or dividing both sides by a negative number.
- Mixing up “and” vs “or” logic: compound inequalities behave very differently depending on which connector is used.
- Dropping one branch of an absolute value case: both the positive and negative case must be solved and combined.
- Including excluded values in rational inequalities: any x-value that makes a denominator zero can never be part of the solution.