Linear Inequality Calculator: How to Solve & Graph Inequalities Step by Step
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
A linear inequality calculator solves expressions like 3x + 5 < 14 in seconds, showing you the exact solution, interval notation, and a number line graph — all with step-by-step work. Whether you are checking homework, preparing for a test, or solving real-world constraint problems, this tool replaces tedious manual algebra with instant, verified results.
In this guide, you will learn the rules for solving linear inequalities by hand, see fully worked examples with real numbers, and understand how to read interval notation and number line graphs. Every concept is demonstrated with the same approach your textbook uses, so the steps transfer directly to paper-and-pencil work.
What Is a Linear Inequality?
A linear inequality is a mathematical statement that compares a linear expression to a value using one of four operators: < (less than), ≤ (less than or equal to), > (greater than), or ≥ (greater than or equal to). The general form is:
ax + b < c or ax + b ≤ c or ax + b > c or ax + b ≥ c
Unlike an equation (which has one answer), an inequality has infinitely many solutions — an entire range of numbers that satisfy the condition. For example, x > 3 means every number larger than 3 works: 3.001, 4, 100, and so on. This is why we express solutions as intervals rather than single values.
Linear inequalities appear constantly in real life. Budget constraints ("spend less than $500"), minimum grade requirements ("score at least 70%"), and speed limits ("drive no more than 65 mph") are all inequality problems. Our inequality calculator handles more complex types including quadratic and absolute value inequalities, while this page focuses specifically on linear (first-degree) ones.
How to Solve a Linear Inequality (The Method)
Solving a linear inequality follows the same steps as solving a linear equation with one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign. Here is the step-by-step method:
- Isolate the variable term — subtract or add the constant from both sides to get the ax term alone
- Divide by the coefficient — divide both sides by the coefficient of x
- Flip the sign if dividing by a negative — this is the rule most students forget, and it changes the entire answer
- Write the solution — express in inequality notation, interval notation, or on a number line
The flipping rule exists because multiplying by a negative reverses the order of numbers. Since -1 × 3 = -3 and -1 × 5 = -5, the larger original number (5) becomes the smaller result (-5). The inequality direction must reverse to stay true.
Worked Examples with Step-by-Step Solutions
Example 1: Simple inequality (no sign flip)
Solve 3x + 5 < 14
- Subtract 5 from both sides: 3x < 9
- Divide both sides by 3: x < 3
Solution: x < 3, or in interval notation: (-∞, 3). The number line shows an open circle at 3 with shading to the left, meaning 3 itself is not included.
Example 2: Negative coefficient (sign flip required)
Solve -2x + 1 ≥ 7
- Subtract 1 from both sides: -2x ≥ 6
- Divide by -2 and flip ≥ to ≤: x ≤ -3
Solution: x ≤ -3, or (-∞, -3]. The square bracket at -3 means -3 is included. This is the example that trips up most students — forgetting to flip ≥ to ≤ gives the wrong answer x ≥ -3, which is the opposite half of the number line.
Example 3: Variables on both sides
Solve 5 - x ≤ 3x + 1
- Subtract 3x from both sides: 5 - 4x ≤ 1
- Subtract 5 from both sides: -4x ≤ -4
- Divide by -4 and flip ≤ to ≥: x ≥ 1
Solution: x ≥ 1, or [1, ∞). You can verify by testing x = 0 (should fail): 5 - 0 = 5 ≤ 3(0) + 1 = 1? No, 5 is not ≤ 1. Testing x = 2 (should pass): 5 - 2 = 3 ≤ 3(2) + 1 = 7? Yes, 3 ≤ 7. ✓
Interval Notation and Number Line Graphs
Interval notation is a compact way to write the solution set. There are two symbols to know: parentheses ( ) mean "not included" (open boundary), and square brackets [ ] mean "included" (closed boundary). Infinity always uses parentheses because you cannot reach infinity.
| Inequality | Interval Notation | Number Line | Boundary |
|---|---|---|---|
| x < a | (-∞, a) | Arrow left from open circle | Not included |
| x ≤ a | (-∞, a] | Arrow left from filled circle | Included |
| x > a | (a, ∞) | Arrow right from open circle | Not included |
| x ≥ a | [a, ∞) | Arrow right from filled circle | Included |
Our calculator above generates both the interval notation and a visual number line graph automatically. For more complex interval problems, you can also use our interval notation calculator which handles unions and intersections of multiple intervals.
Common Mistakes to Avoid
- Forgetting to flip the sign when dividing by a negative. This is the #1 error. Dividing -2x ≥ 6 by -2 gives x ≤ -3, not x ≥ -3. The wrong answer represents the opposite half of the number line — every single value is wrong.
- Confusing open and closed circles. Strict inequalities (<, >) use open circles (boundary excluded). Non-strict (≤, ≥) use closed circles (boundary included). Writing x < 5 with a closed circle at 5 means 5 is in the solution — but plugging in x = 5 gives equality, not "less than."
- Distributing a negative incorrectly. In -2(x - 3) > 4, students often write -2x - 6 > 4 instead of -2x + 6 > 4. The negative must distribute to both terms inside the parentheses: -2 × (-3) = +6.
- Swapping sides without flipping. If 5 < x, that is the same as x > 5 — reading the inequality from the variable's perspective. When you swap sides, the operator must also flip.
Linear Inequality vs. Linear Equation
| Feature | Linear Equation | Linear Inequality |
|---|---|---|
| Symbol | = | <, ≤, >, ≥ |
| Solution count | Exactly one value | Infinite values (a range) |
| Graph | A single point on the line | A ray or segment on the line |
| Multiply by negative | No special rule | Must flip the inequality sign |
| Notation | x = 5 | x > 5 or (5, ∞) |
If you need to solve equations rather than inequalities, our linear equation calculator handles standard form, slope-intercept form, and two-variable systems. For finding the exact value of x in an equation, try the solve for x calculator.
Tips for Solving Inequalities Accurately
- Always verify by substitution. Pick a number from your solution set and plug it back into the original inequality. If x > 3, test x = 4. If the original inequality holds true, you likely solved correctly.
- Test the boundary point. If your answer is x < 5, plug in x = 5. It should make the two sides equal (not satisfy the strict inequality). This confirms you found the right boundary.
- Test a number from the excluded region. If x > 3, test x = 0. The original inequality should be false. If it is true, you made an error — likely a missed sign flip.
- Simplify both sides first. Before isolating x, combine like terms and distribute. Errors in arithmetic cascade into wrong answers.
- Keep the variable on the left. While mathematically equivalent, writing x > 5 is clearer than 5 < x for most students and reduces reading errors.
When to Use This Calculator
- Homework verification — solve the inequality by hand first, then use the calculator to check your work and compare each step
- Test preparation — practice with the example buttons to build fluency with all four inequality types before an exam
- Interval notation practice — the calculator shows inequality, interval, and set-builder notation side by side so you can learn to convert between them
- Understanding the sign-flip rule — try examples with negative coefficients and watch the step-by-step solution show exactly when and why the sign flips
For compound inequalities like 2 < 3x + 1 < 10, use our compound inequality calculator which handles AND/OR logic and double inequalities. For systems of multiple inequalities with two variables, a graphing approach with shaded regions works better than a number line.
