Solve an inequality and graph its solution
Enter one linear inequality. Use x, decimals, spaces, and comparison symbols. Examples: 2x - 3 <= x + 7, -3x > 12, or x != 4.
Common inequality conversions
| Inequality | Solution | Interval notation | Number-line mark |
|---|---|---|---|
| 3x - 6 < 9 | x < 5 | (-∞, 5) | Open circle at 5, shade left |
| -2x + 4 ≥ 10 | x ≤ -3 | (-∞, -3] | Closed circle at -3, shade left |
| x + 8 > 12 | x > 4 | (4, ∞) | Open circle at 4, shade right |
| 5x = 20 | x = 4 | {4} | Closed point at 4 |
| 7x - 2 != 19 | x ≠ 3 | (-∞, 3) ∪ (3, ∞) | Open point at 3, shade both sides |
Convert both sides into one linear expression
Starting form: ax + b ⋚ cx + d
Move terms: (a - c)x ⋚ d - b
Boundary: x = (d - b) ÷ (a - c), when a - c is not zero.
When you divide by a negative value, reverse the comparison direction. For strict symbols, use parentheses and open circles. For inclusive symbols, use brackets and closed circles.
How to use this calculator
- Type a linear inequality with one variable. Keep the same variable on both sides.
- Choose the number of decimal places for boundary values and graph labels.
- Leave graph half-span blank for automatic scaling, or enter a positive viewing distance.
- Select Solve Inequality to see the answer above the input form.
- Review the interval, endpoint style, calculation steps, and shaded number line.
- Use the CSV button for a table export or the PDF button for a print-ready result.
Understanding Inequalities on a Number Line
An inequality compares values instead of declaring them equal. It tells you which values make a statement true. A number line turns that statement into a visual answer. The boundary value marks the point where the comparison changes. A hollow circle means the boundary is excluded. A filled circle means it is included. Shading shows every value that belongs to the solution set.
Why the Comparison Symbol Matters
The symbols less than and greater than create open endpoints. The symbols less than or equal to and greater than or equal to create closed endpoints. When the variable coefficient is negative, dividing changes the inequality direction. For example, if minus two x is greater than eight, dividing by minus two gives x less than minus four. The endpoint stays open because the original symbol was greater than, not greater than or equal to.
Reading Interval Notation
Interval notation provides a compact written form for a number-line answer. Parentheses show excluded endpoints. Square brackets show included endpoints. Infinity always uses a parenthesis because infinity is not a reachable number. Therefore, x greater than three becomes open parenthesis three, infinity parenthesis. In contrast, x less than or equal to five becomes open parenthesis negative infinity, five square bracket. A single exact answer can be written with braces. A statement such as x equals two has the solution set brace two brace.
Solving Expressions on Both Sides
Some inequalities place the variable on both sides. Start by moving variable terms to one side. Move constant terms to the other side. Suppose two x plus three is less than x plus seven. Subtract x from both sides. Then subtract three from both sides. The result is x less than four. The calculator follows the same balance rule. It combines like terms, identifies the boundary, and selects the correct endpoint style after checking the final comparison sign.
Special Results and Practical Checks
Not every input produces a single ray. A true constant comparison can produce all real numbers. A false constant comparison produces no solution. Equality produces one closed point. Not-equal produces two shaded directions with an open point in the middle. Check answers by testing a value from the shaded region. Test another value from the unshaded region. The first should satisfy the original inequality. The second should fail. This simple check catches sign mistakes and helps confirm the graph matches the algebra.
Using the Graph in Everyday Work
Number-line inequalities appear in limits, budgets, measurements, schedules, and safety ranges. A minimum temperature becomes a greater-than-or-equal-to statement. A maximum file size becomes a less-than-or-equal-to statement. A permitted operating range may require two separate inequalities. The graph makes these limits easier to communicate. Choose sensible rounding when estimates are enough. Keep the original units beside your work. Clear endpoints make inequality results easy to share and reuse for future reference.
Helpful answers
1. What does an open circle mean?
An open circle means the boundary value is excluded. Use it with less than, greater than, or not-equal comparisons. The shaded direction still shows values that satisfy the inequality.
2. What does a closed circle mean?
A closed circle includes the boundary value. Use it with less than or equal to, greater than or equal to, and equality comparisons.
3. Why does the symbol reverse after dividing?
Dividing or multiplying both sides by a negative number reverses order on the number line. Therefore, less than becomes greater than, and less than or equal to becomes greater than or equal to.
4. Can I enter decimals?
Yes. Use decimal coefficients or constants, such as 1.5x - 2.25 >= 4. The selected decimal setting controls how displayed results are rounded.
5. Can the variable appear on both sides?
Yes. Enter expressions such as 2x + 3 <= x + 7. The calculator combines matching variable terms before locating the boundary value.
6. What interval notation uses infinity?
Infinity always uses parentheses. It is not a number that can be included. For example, x greater than four is written as (4, ∞).
7. What happens when the variable cancels?
The remaining statement may be always true or always false. A true statement means all real numbers work. A false statement means there is no solution.
8. How is equality shown on the number line?
Equality has one exact solution, so it uses one closed point. Its set notation uses braces, such as {2}, rather than an interval.
9. How is not-equal shown?
Not-equal excludes one boundary but includes values on both sides. The graph has an open point and two shaded rays. Its notation joins two open intervals.
10. Does graph half-span change the answer?
No. It only changes the visible scale around the boundary. The solved inequality, interval notation, and endpoint inclusion remain exactly the same.
11. How can I check a solved inequality?
Test one value inside the shaded region and one outside it. Accurate endpoints make every number-line decision easier to trust.