Calculator Inputs
Example Data Table
| Case | Input statement | Expected interval result | Reason |
|---|---|---|---|
| Linear | 2x - 6 ≤ 0 | (-∞, 3] | Divide by 2 without flipping the sign. |
| Quadratic | x² - 5x + 6 > 0 | (-∞, 2) ∪ (3, ∞) | The parabola is positive outside its real roots. |
| Absolute | |2x - 3| ≤ 5 | [-1, 4] | Convert to -5 ≤ 2x - 3 ≤ 5. |
| Compound | -3 ≤ 2x + 1 ≤ 7 | [-2, 3] | Intersect the two resulting linear ranges. |
| Equality | x² - 4x + 3 = 0 | {1} ∪ {3} | Equality keeps only the exact real roots. |
Formula Used
1) Linear form
For ax + b ? 0, move b first, then divide by a. Reverse the comparison whenever a is negative.
2) Quadratic form
For ax² + bx + c ? 0, compute Δ = b² - 4ac. Real roots split the number line into sign intervals. Test each interval.
3) Absolute value form
For |ax + b| ≤ c, solve -c ≤ ax + b ≤ c. For |ax + b| ≥ c, solve ax + b ≥ c or ax + b ≤ -c.
4) Compound form
For L ≤ ax + b ≤ U, solve both sides separately, then intersect the two solution sets.
How to Use This Calculator
- Choose the solving mode that matches your inequality type.
- Enter coefficients, constants, target values, or compound limits.
- Pick the comparison operator, including equality when needed.
- Press Solve inequality to display the result above the form.
- Read the interval notation, boundary values, and solving notes.
- Use the graph to verify crossings, shaded intervals, and solution regions.
- Download CSV or PDF reports for revision, teaching, or documentation.
FAQs
1) What does the shaded graph region mean?
The shaded strip highlights x-values that satisfy the entered condition. Use it with the curve and boundary markers to confirm the final interval notation visually.
2) Why does the sign flip when dividing?
Dividing or multiplying an inequality by a negative number reverses order on the number line. That is why less-than becomes greater-than, and vice versa.
3) How are repeated quadratic roots treated?
A repeated root touches the x-axis without crossing it. The point may be included for non-strict comparisons, but sign behavior on both sides stays the same.
4) Can this tool solve equality too?
Yes. Choose the equals operator to solve exact real roots or exact boundary points. The output then appears as one point or a finite set of points.
5) What happens when no real solution exists?
The calculator returns the empty set symbol, ∅. That means no real x-value satisfies the entered condition under the real-number domain used here.
6) Why are absolute inequalities split into cases?
Absolute value measures distance from zero. Because distance can match positive and negative directions, the inequality becomes two linked linear conditions or two separate branches.
7) Does the calculator assume real numbers only?
Yes. Interval notation, number-line shading, and inequality order all assume real numbers. Complex-number solutions are not ordered and are not shown here.
8) When should I use interval notation versus points?
Use intervals when infinitely many x-values satisfy the condition. Use points or finite sets when equality or boundary-only cases produce isolated real solutions.