Solve a linear inequality
Use the form a·x + b comparison c·x + d. The calculator isolates one variable and keeps the correct comparison direction.
Formula used
Move variable terms to one side. Move constants to the other side. Divide by the remaining coefficient. Reverse the comparison only when dividing by a negative number.
How to use this calculator
- Enter the coefficient and constant from the left expression.
- Choose the correct comparison symbol from the list.
- Enter the coefficient and constant from the right expression.
- Set a variable name and select the desired decimal precision.
- Select Solve inequality to view the result above this form.
- Download the displayed work as CSV or PDF when needed.
Example calculations
| Input statement | Normalized form | Solution | Interval |
|---|---|---|---|
| 3x + 7 ≤ x + 15 | 2x ≤ 8 | x ≤ 4 | (-∞, 4] |
| -2x + 5 > 11 | -2x > 6 | x < -3 | (-∞, -3) |
| 4x - 8 ≠ 4x + 2 | 0 ≠ 10 | All real numbers | (-∞, ∞) |
Understanding inequality solutions
What the result means
An inequality describes a range of possible values. The calculator returns the range in solved form and interval notation. A closed bracket includes the endpoint. A parenthesis excludes it. Equality produces one exact value. A not-equal comparison excludes one boundary value.
Why the sign can reverse
Multiplying or dividing by a negative number changes order on the number line. For that reason, less than becomes greater than. Greater than becomes less than. This rule protects the meaning of the original comparison. The result panel tells you when the reversal happened.
Collecting terms correctly
Start by moving variable terms together. Then move numeric terms together. This creates one coefficient beside the variable. Divide by that coefficient to finish. The calculator uses the compact transformation (a − c)x comparison (d − b). It shows the normalized statement before giving the final set.
Handling special cases
Sometimes the variable terms cancel. The statement then becomes a comparison between two constants. A true comparison means every real number works. A false comparison means no real number works. This is why an inequality does not always end with a single boundary point.
Reading interval notation
Interval notation is useful for graphing and checking answers. The symbol infinity always uses a parenthesis. A square bracket is only used with a real endpoint that is included. The union symbol joins separated ranges. It commonly appears when the comparison is not equal to.
Checking a solution
Choose a test value from the reported range. Substitute it into the original statement. Both sides must preserve the selected comparison. For a strict boundary, test a nearby value instead. For example, x < 4 accepts 3.9 but rejects 4. This habit catches copied signs and data-entry errors before reports are shared.
Graphing the answer
A number line gives another useful view. Place the boundary point first. Draw an open circle for a strict comparison. Draw a filled circle for an inclusive comparison. Shade left for less-than results. Shade right for greater-than results. With not equal to, leave the boundary open and shade both directions. The interval shown here matches that picture.
Practical uses
Inequalities appear in budget limits, tolerances, capacity checks, and target ranges. A shipping rule might require weight under a limit. A production rule might require temperature at least a threshold. The same structure applies whenever one quantity must stay above, below, or outside a value. Confirm the units before interpreting a calculated boundary.
Rounding and precision
Rounding changes how a displayed endpoint looks. It does not change the underlying computation. Use more decimal places when measurements are close to a limit. Avoid rounding intermediate manual steps too early. When a boundary has repeating decimals, retain enough digits for the decision. The selected precision controls presentation and exports carefully.
Use results with care
Enter values exactly as written in the original statement. Keep negative signs with the correct coefficient or constant. Increase decimal places for measurements that need more detail. Check the displayed original statement before using the solution. Exported files preserve the main result and the working steps.
Frequently asked questions
1. What kinds of statements can this calculator solve?
It solves single-variable linear comparisons in the form a·x + b compared with c·x + d. You can use less than, less than or equal, greater than, greater than or equal, equal to, and not equal to.
2. Why did the comparison sign reverse?
The sign reverses when the final variable coefficient is negative. Dividing an inequality by a negative value reverses its order. Equality and not-equal comparisons do not reverse under this rule.
3. Can I use decimal numbers?
Yes. Enter decimals for any coefficient or constant. Choose the decimal-place setting that best matches your data. The calculation keeps full numeric precision internally before formatting the shown result.
4. What does all real numbers mean?
It means every possible real value satisfies the comparison. This usually happens after the variable terms cancel and the remaining constant comparison is true.
5. What does no solution mean?
It means no real value can make the comparison true. This usually happens after the variable terms cancel and the remaining constant comparison is false.
6. How are brackets and parentheses chosen?
A square bracket includes the boundary value. A parenthesis excludes it. Strict comparisons use parentheses. Inclusive comparisons use square brackets at the finite endpoint.
7. Can I solve an equation instead?
Yes. Select the equal-to option. The result becomes the value that makes both sides equal. The interval display uses braces for that single solution.
8. What happens with not equal to?
The calculator returns every real number except the boundary value. Its interval notation uses two open intervals joined by a union symbol.
9. Can I change the variable name?
Yes. Use a letter, then optionally add letters, numbers, or underscores. The chosen name appears in the preview, result, steps, and downloads.
10. Are downloaded files based on my current values?
Yes. After solving, the CSV and PDF buttons export the current expression, normalized comparison, final solution, interval notation, interpretation, and calculation steps.
11. Should I verify the answer manually?
For important work, substitute a test value from the reported range into the original statement. This quick check confirms the direction, endpoint choice, and entered signs.