Compound Linear Inequality Calculator

Enter two inequalities and combine them precisely. View interval notation, set builder form, and graphs. Test endpoints, compare unions, and verify every algebraic step.

Enter Calculator Values

General Settings

Use the form ax + b relation cx + d for each inequality.

Inequality 1

Inequality 2

Example Data Table

Example Connector Reduced forms Final solution Interval notation
2x + 3 < 11 AND 3x - 2 ≥ 4 AND x < 4, x ≥ 2 2 ≤ x < 4 [2, 4)
4x - 1 ≤ 7 OR x + 5 > 12 OR x ≤ 2, x > 7 x ≤ 2 or x > 7 (-∞, 2] ∪ (7, ∞)
-2x + 6 > 0 AND 5x + 10 ≤ 20 AND x < 3, x ≤ 2 x ≤ 2 (-∞, 2]
3x + 9 < 3x + 1 OR 2x - 4 ≥ 8 OR No solution, x ≥ 6 x ≥ 6 [6, ∞)

Formula Used

Each inequality is solved in standard algebraic form:

ax + b relation cx + d (a - c)x relation (d - b) x relation (d - b) / (a - c)

After moving all variable terms to one side and constants to the other, divide by the remaining coefficient. If that coefficient is negative, reverse the inequality sign.

Compound rules:

  • AND means take the intersection of both solution sets.
  • OR means take the union of both solution sets.

How to Use This Calculator

  1. Enter coefficients and constants for the first inequality.
  2. Select the correct inequality sign for that statement.
  3. Enter the second inequality in the same form.
  4. Choose AND for overlap or OR for either statement being true.
  5. Select the output precision you want.
  6. Press Solve Inequality to see algebra steps, interval notation, set builder form, and the graph.
  7. Use the CSV or PDF buttons to export the result summary.

FAQs

1. What is a compound linear inequality?

A compound linear inequality combines two linear inequalities using AND or OR. AND asks for overlap. OR asks for values that satisfy at least one statement.

2. What does AND mean in the result?

AND means both inequalities must be true at the same time. The calculator finds the intersection, which is the shared part of both solution sets.

3. What does OR mean in the result?

OR means either inequality may be true. The calculator forms the union, which includes every value that belongs to at least one solution set.

4. Why does the inequality sign sometimes flip?

The sign flips only when dividing or multiplying by a negative number. This preserves the correct order of values on the number line.

5. What do open and closed circles mean on the graph?

An open circle means the endpoint is excluded. A closed circle means the endpoint is included in the solution set.

6. Can the calculator return no solution?

Yes. This happens when the two inequalities do not overlap under AND, or when a single inequality is impossible after simplification.

7. Can the calculator return all real numbers?

Yes. If an inequality simplifies to a statement that is always true, its solution can be all real numbers. The final compound result may also be all real numbers.

8. Why are interval notation and set builder notation both shown?

They present the same solution in different formats. Interval notation is compact, while set builder notation states the logical condition directly.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.