Solve any linear inequality instantly — get step-by-step solutions, number line graphs, and interval notation. Free, no sign-up required.
Why this page ranks for linear intent
Linear Solver
Built for one-variable linear inequalities with instant steps, a number line, interval notation, and linear two-variable graphing when your input includes y.
Supported Input Styles
2x+3>7 becomes a linear one-variable inequality.5-x<=2 keeps the negative coefficient logic visible.3(x+1)>=-8 expands parentheses before solving.x/2+3>1 treats the fraction as part of the linear expression.2x+3y<6 switches to coordinate-plane graphing automatically.Math Keyboard
Tap symbols, numbers, or actions for fast linear-inequality input.
Result
The linear inequality isolates x > 2.
Step 1
Move all terms to one side
Rewrite the inequality so the right-hand side is zero.
Before
After
Step 2
Isolate x
Add or subtract the constant term so that the x-term stands alone.
Before
After
Step 3
Divide by the coefficient of x
Dividing both sides by a positive number keeps the inequality direction the same.
Before
After
Recent History
Saved locally in this browser so you can jump back into recent linear problems without any backend.
Calculator Types
Focused on linear steps, graphing, interval notation, and direct verification.
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.
Solve rational inequalities with steps, excluded values, sign charts, 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
The fastest way to use the linear inequality calculator is to type the inequality exactly as you would write it in class. You can enter forms such as 2x + 3 > 7, 5 - x <= 2, 3(x + 2) >= 12, or x/2 - 3 > 1. The parser accepts common keyboard input, so you do not need special math markup just to get started.
After you enter a problem, the page keeps the result area tightly focused on the four views that matter most for linear work: algebraic steps, the graph, interval notation, and verification. That structure is intentional. Many calculator pages stop after they isolate x, but students usually need one more layer of interpretation. They want to know what the answer means on a number line and whether a test value really works.
If you are checking homework, begin with the Steps tab and compare each transformation to your own work. If you are learning the topic for the first time, switch to the Graph and Interval Notation tabs right after that. Seeing the algebra and the visual answer together is the quickest way to understand why x > 2 becomes an open endpoint with shading to the right, while x <= 5 becomes a closed endpoint with shading to the left.
Enter a linear inequality such as 2x + 3 > 7, 5 - x <= 2, or 2x + 3y < 6.
Review the step cards to see how the variable is isolated and when the inequality sign stays the same or flips.
Open the graph, interval notation, and verify tabs to confirm the answer from several angles.
A linear inequality compares two first-degree expressions. Instead of asking when two expressions are equal, it asks when one side is greater than, less than, or equal to the other in an inequality sense. That is why a problem like 2x + 3 > 7 usually has many correct values. Every real number larger than 2 works, so the answer is a set of values, not a single solution.
This is the key distinction between a linear equation and a linear inequality. An equation such as 2x + 3 = 7 has one exact answer, x = 2. The related inequality 2x + 3 > 7 has an entire ray of answers, x > 2. The variable still has degree one, the algebra rules are still familiar, but the final interpretation is different because the result describes a range.
You will most often see four symbols in linear inequality work: >, <, >=, and <=. Strict symbols, > and <, exclude the boundary value. Inclusive symbols, >= and <=, include the boundary value. That inclusion rule shows up everywhere: in the final isolated statement, in interval notation, and on the graph where the endpoint changes from an open circle to a closed circle.
Equation example
Inequality example
| Comparison point | Linear equation | Linear inequality |
|---|---|---|
| Goal | Find the exact value that makes both sides equal. | Find every value that keeps the comparison true. |
| Typical answer | A single value such as x = 2. | A range such as x > 2 or -1 <= x < 4. |
| Graph | One point on a number line. | A shaded ray or interval. |
| Notation | Usually a single algebraic statement. | Often written in inequality, interval, and set notation. |
The algebra behind linear inequalities is familiar on purpose: combine like terms, isolate the variable, and read the final statement carefully. The difference is in interpretation. Every move must preserve the truth of the comparison, and one rule matters more than any other rule on the page: dividing by a negative number reverses the inequality symbol.
Adding or subtracting the same value keeps the balance of the comparison. The inequality direction does not change because you are shifting both sides by the same amount.
Example
When the number is positive, the order of values stays the same. Dividing 6 > 2 by 2 gives 3 > 1, so the inequality symbol remains unchanged.
Example
This is the most common source of mistakes. Negative multiplication reverses order, so the symbol must flip direction. If -4x < 8, then x > -2 after dividing by -4.
Example
If x appears on both sides, gather the variable terms first. This makes the structure easier to read and reveals whether the remaining coefficient of x is positive or negative.
Example
A distributed form is much easier to solve correctly. Expand, combine like terms, then continue with the usual add, subtract, multiply, or divide steps.
Example
If you divide by a negative coefficient and keep the original symbol, you reverse the meaning of the answer. Always pause before the last step when the coefficient of x is negative.
Strict inequalities use open circles and parentheses. Inclusive inequalities use closed circles and brackets. Matching these consistently keeps the graph and interval notation aligned.
Students often change +3 into +3 again instead of -3 when moving it across the inequality. The cleaner habit is to add or subtract the same value on both sides rather than relying on an informal move-term shortcut.
A missed negative sign during distribution can break the whole solution. Expand slowly, then combine like terms before you isolate the variable.
The six examples below were chosen to cover the most common classroom patterns: basic isolation, a negative coefficient, parentheses, a forced sign flip, fractions, and variables on both sides. If you can explain why each final graph looks the way it does, you understand the topic at a usable level.
This is the simplest linear-inequality pattern. The variable is already on one side, so you only need one subtraction step and one division step.
Subtract 3 from both sides
Before
After
Divide both sides by 2
Before
After
Answer
Interval
Set notation
Number line
This example teaches an important idea: the sign does not flip when you subtract 5, but it does flip when you divide by -1.
Subtract 5 from both sides
Before
After
Divide both sides by -1
Before
After
Answer
Interval
Set notation
Number line
Parentheses hide the linear structure. Once you distribute, the inequality becomes a familiar two-step problem.
Distribute the 3
Before
After
Subtract 6 from both sides
Before
After
Divide both sides by 3
Before
After
Answer
Interval
Set notation
Number line
This example is a classic check for sign-flip discipline. The final step is simple, but it is also where most errors happen.
Subtract 1 from both sides
Before
After
Divide both sides by -4
Before
After
Answer
Interval
Set notation
Number line
Fractions are easier when you isolate the fractional term first. Because you multiply by a positive number, the sign stays exactly the same.
Add 3 to both sides
Before
After
Multiply both sides by 2
Before
After
Answer
Interval
Set notation
Number line
When x appears on both sides, your first goal is to collect the variable terms into one place. Only then can you see the final direction of the answer.
Subtract 2x from both sides
Before
After
Add 4 to both sides
Before
After
Rewrite in standard variable-first form
Before
After
Answer
Interval
Set notation
Number line
A number line translates algebra into a picture. Once you isolate x, the graph answers two quick questions: where is the boundary, and which side should be shaded? For x > 2, the boundary is 2 and the shading extends to the right because larger values satisfy the inequality. For x < 2, the boundary is still 2, but the shading moves left because smaller values work instead.
The endpoint style matters just as much as the direction. Open circles match strict symbols, > and <, because the boundary itself does not belong to the solution. Closed circles match inclusive symbols, >= and <=, because the endpoint is part of the answer. If you can read a number line correctly, interval notation becomes much more intuitive because you can see why (2, infinity) excludes 2 while [2, infinity) includes it.
Strict inequality
Use an open circle when the endpoint is excluded, as in x > 2 or x < -1.
Inclusive inequality
Use a closed circle when the endpoint is included, as in x >= 2 or x <= -1.
Interval notation is the compact form of a graph. Parentheses mean an endpoint is excluded, and brackets mean an endpoint is included. Infinity is never written with a bracket because infinity is not a reachable endpoint on the real line.
A good habit is to read interval notation out loud. For example, [3, infinity) means every real number from 3 upward, with 3 included. That verbal check helps you catch mistakes before you turn in homework or copy the final answer into another system.
| Inequality | Interval notation | Number line | Meaning |
|---|---|---|---|
| Values greater than a, but not including a. | |||
| Values greater than a, including a. | |||
| Values less than a, but not including a. | |||
| Values less than a, including a. |
A linear inequality is a comparison that uses a first-degree expression, such as 2x + 3 > 7 or 5 - x <= 2. Its solution is usually a range of values, not one single number.
A linear equation asks for the exact value that makes two sides equal, while a linear inequality asks for every value that makes one side greater than, less than, greater than or equal to, or less than or equal to the other.
The four core inequality symbols are >, <, >=, and <=. They mean greater than, less than, greater than or equal to, and less than or equal to.
The solution represents every real number that makes the original statement true. That is why answers are often written as intervals, rays, or shaded regions instead of a single point.
Simplify both sides, move variable terms to one side, move constants to the other, and divide by the coefficient of the variable. If that coefficient is negative, reverse the inequality symbol.
You flip the inequality sign only when you multiply or divide both sides by a negative number. Adding or subtracting values does not reverse the inequality.
Collect all x-terms on one side and all constant terms on the other side first. Once the inequality is rewritten as ax > b or ax <= b, divide by a and flip the sign only if a is negative.
Clear the fraction carefully or isolate the fractional term first. As long as you multiply or divide by a positive number, the inequality sign stays the same.
Distribute first so the expression is fully expanded. After that, combine like terms and solve as a standard linear inequality.
Yes. If the variable terms cancel and the remaining statement is false, such as 4 < 1, then no real number satisfies the inequality.
Yes. If the variable terms cancel and the remaining statement is always true, such as 2 <= 5, then every real number is a solution.
Use parentheses for strict inequalities and brackets for inclusive inequalities. For example, x > 2 becomes (2, infinity), while x >= 2 becomes [2, infinity).
Plot the boundary value first, draw an open or closed circle depending on whether the endpoint is excluded or included, and shade left or right depending on the direction of the inequality.
An open circle means the endpoint is not included, so it is used with < or >. A closed circle means the endpoint is included, so it is used with <= or >=.
Set notation describes the rule directly, such as {x | x > -3}. It tells you which values of x belong to the solution set.
Type a linear inequality into the input, tap Solve or press Enter, and then switch between the steps, graph, interval notation, and verify tabs. Example buttons are included if you want to start with a model problem.
Yes. The calculator explains each main algebra move, highlights the final isolated statement, and calls out the sign-flip rule when a negative coefficient is involved.
Yes. If your input contains y, the calculator switches from number-line mode to coordinate-plane mode and shades the correct half-plane.
Yes. The calculator, graph, interval notation, and explanation content are available without a sign-up.
This page is optimized for linear inequalities, including expressions with parentheses, fractions, variables on both sides, and linear two-variable inequalities for graphing.
Zone 6
Once you are comfortable with linear inequalities, the next useful jumps are compound inequalities, absolute-value inequalities, and graphing-focused practice. Those pages extend the same algebra rules into new answer shapes and new visual interpretations.
Compound Inequality Calculator
Solve compound inequalities with interval intersection, union logic, and graph output.
Absolute Value Inequality Calculator
Solve absolute value inequalities with case splitting, interval notation, and step-by-step explanations.
Graphing Inequalities Calculator
Graph linear, quadratic, and absolute value inequalities in two variables with shading, dashed or solid boundaries, and system overlap tools.