Calculator Input
This form solves inequalities written as: left value, symbol, bx + c, symbol, right value.
Example data table
| Input form | Example inequality | Solution | Interval |
|---|---|---|---|
| a < bx + c ≤ d | -3 < 2x + 5 ≤ 17 | -4 < x ≤ 6 | (-4, 6] |
| a ≤ bx + c < d | 1 ≤ 3x - 2 < 10 | 1 ≤ x < 4 | [1, 4) |
| a > bx + c ≥ d | 5 > -2x + 1 ≥ -7 | -2 ≤ x < 3 | [-2, 3) |
| a ≥ bx + c > d | 9 ≥ x + 4 > 1 | -3 < x ≤ 5 | (-3, 5] |
Formula used
General form: a ◁ bx + c ▷ d
Left comparison: a ◁ bx + c → a - c ◁ bx → x swap(◁) (a - c)/b
Right comparison: bx + c ▷ d → bx ▷ d - c → x ▷ (d - c)/b
Important rule: If b is negative, reverse each inequality when dividing by b.
After solving both single inequalities, intersect their solution sets.
That intersection gives the final compound statement, interval notation, and graph range.
How to use this calculator
Enter the left number, pick the left inequality sign, then enter the coefficient and constant for the middle expression.
Choose the right inequality sign and enter the right number. Add an optional test value if you want a quick check.
Select the number of decimal places and press the solve button. The page shows the solved interval above the form.
Use the CSV and PDF buttons to save the result for homework, notes, or teaching materials.
FAQs
1. What is a double inequality?
A double inequality compares one expression to two bounds at the same time. It tells you which values of x make both comparisons true together.
2. Why do inequality signs reverse with negative division?
Multiplying or dividing an inequality by a negative number reverses order on the number line. A larger number becomes smaller after the sign change, so the symbol must flip.
3. What does interval notation show?
Interval notation gives the final answer as a range. Parentheses mean an endpoint is excluded, while brackets mean the endpoint is included.
4. Can the solution be empty?
Yes. If the two solved bounds do not overlap, no real number can satisfy both comparisons at once. The result is an empty set.
5. Can the answer be all real numbers?
Yes. This can happen when the variable term disappears and the remaining statement is always true, or when the solved comparisons place no real restriction on x.
6. What is the optional test value used for?
It checks one chosen x value in the original inequality. This helps confirm whether a specific number belongs inside the final solution interval.
7. Does this work with decimals and negative numbers?
Yes. The calculator accepts positive values, negative values, and decimals for all numeric inputs, then formats the result using your chosen precision.
8. Why are there two solved single inequalities?
A double inequality is split into left and right comparisons. Each is solved separately, then the two results are intersected to produce the final answer.