Compound Inequality Calculator

Compound Inequality Calculator

Understanding Compound Inequalities

A compound inequality combines two or more simple inequalities with either the word "AND" or the word "OR." These inequalities are used to describe a range of values that a variable can take.

"AND" Compound Inequalities (Intersection)

When two inequalities are joined by "AND," the solution set includes only the values that satisfy *both* inequalities simultaneously. This is often referred to as the intersection of the two individual solution sets. Graphically, it represents the overlapping region of the two inequalities.

Example: If you have x > 3 AND x ≤ 7, the solution includes all numbers greater than 3 but less than or equal to 7. This can be written as 3 < x ≤ 7 in inequality form, or (3, 7] in interval notation.

• If the solution sets of the individual inequalities do not overlap, there is "No Solution."
• If one inequality's solution set is entirely contained within the other, the solution is the smaller (more restrictive) set.

"OR" Compound Inequalities (Union)

When two inequalities are joined by "OR," the solution set includes any value that satisfies *at least one* of the inequalities. This is known as the union of the two individual solution sets. Graphically, it represents all regions covered by either inequality.

Example: If you have x < 2 OR x ≥ 5, the solution includes all numbers less than 2, as well as all numbers greater than or equal to 5. These are two separate, disjoint intervals. In inequality form, it remains x < 2 OR x ≥ 5, and in interval notation, it's (-∞, 2) ∪ [5, ∞).

• If the solution sets overlap or touch, the "OR" condition often results in a single, larger interval. For instance, x > 5 OR x < 10 covers all real numbers, as every number is either greater than 5 or less than 10 (or both).
• If one inequality's solution set completely contains the other, the solution is the larger (less restrictive) set.

How to Use the Calculator

1. Select First Inequality Operator: Choose <, ≤, >, or ≥ for the first inequality (e.g., x > value1).
2. Enter First Inequality Value: Input the numerical value for the first inequality.
3. Choose Logical Connector: Select "AND" or "OR" to combine the two inequalities.
4. Select Second Inequality Operator: Choose the operator for the second inequality (e.g., x ≤ value2).
5. Enter Second Inequality Value: Input the numerical value for the second inequality.
6. Click "Calculate Solution": The calculator will display the combined solution in both inequality form and interval notation.

Examples:

• Example 1 (AND): x > 3 AND x ≤ 7
  • First Operator: >, Value: 3
  • Connector: AND
  • Second Operator: ≤, Value: 7
  • Result: 3 < x ≤ 7, Interval: (3, 7]
• Example 2 (OR – Disjoint): x < 2 OR x ≥ 5
  • First Operator: <, Value: 2
  • Connector: OR
  • Second Operator: ≥, Value: 5
  • Result: x < 2 OR x ≥ 5, Interval: (-∞, 2) ∪ [5, ∞)
• Example 3 (AND – No Solution): x ≥ 0 AND x < 0
  • First Operator: ≥, Value: 0
  • Connector: AND
  • Second Operator: <, Value: 0
  • Result: No Solution, Interval: {}
• Example 4 (OR – All Real Numbers): x > 5 OR x < 10
  • First Operator: >, Value: 5
  • Connector: OR
  • Second Operator: <, Value: 10
  • Result: All Real Numbers, Interval: (-∞, ∞)
• Example 5 (AND – Overlapping): x > 5 AND x > 2
  • First Operator: >, Value: 5
  • Connector: AND
  • Second Operator: >, Value: 2
  • Result: x > 5, Interval: (5, ∞)

Error:

Solution:

Inequality Form:

Interval Notation: