Calculator Form
Enter two linear inequalities in x. Each row follows the pattern ax + b relation cx + d.
Example Data Table
| Case | Inequality 1 | Inequality 2 | Shared solution | Interval notation |
|---|---|---|---|---|
| Example 1 | 3x - 1 > 5 | 2x + 4 ≤ 16 | 2 < x ≤ 6 | (2, 6] |
| Example 2 | 4x + 8 ≥ 0 | x - 7 < 2 | -2 ≤ x < 9 | [-2, 9) |
| Example 3 | 5x - 10 < 15 | 2x + 1 > 9 | 4 < x < 5 | (4, 5) |
| Example 4 | x + 3 ≤ 2 | x - 1 ≥ 4 | No solution | ∅ |
Formula Used
Start with each inequality:
a1x + b1 relation c1x + d1
a2x + b2 relation c2x + d2
Rearrange each one: (a - c)x relation (d - b)
Divide by (a - c): x relation (d - b)/(a - c)
Important rule: reverse the inequality sign when dividing by a negative number, then intersect both resulting intervals.
How to Use This Calculator
- Enter the coefficients for the first inequality.
- Select the correct relation sign for that row.
- Enter the right-side coefficients for the first row.
- Repeat the same process for the second inequality.
- Choose the decimal precision you want in the output.
- Press the solve button to see the shared interval.
- Review the transformed steps, interval notation, and graph.
- Download the result table as CSV or PDF if needed.
Frequently Asked Questions
1) What is a simultaneous inequality?
It is a pair or group of inequalities solved together. The answer is the overlap of all valid values, not the result from only one inequality.
2) Why do we intersect intervals?
Each inequality gives a possible range. A simultaneous problem needs values that satisfy every condition, so the final answer is the common intersection only.
3) When does the inequality sign reverse?
It reverses when you divide or multiply both sides by a negative number. This step is essential and often decides whether the final interval is correct.
4) What do open and closed endpoints mean?
An open endpoint means the boundary value is excluded. A closed endpoint means the boundary value is included in the solution set.
5) Can the final answer be empty?
Yes. If the two inequalities do not overlap at all, there is no shared solution and the answer becomes the empty set.
6) Can the answer be all real numbers?
Yes. This happens when both inequalities are always true or when one gives all real numbers and the other does not restrict x further.
7) Does this handle decimals and negatives?
Yes. You can enter decimal coefficients, negative coefficients, and negative constants. The calculator formats the result using your chosen precision.
8) What does the graph show?
The graph shows the number line for x. The highlighted segment marks the values that satisfy both inequalities together, including open or closed endpoints.