The primary difference between "or" and "and" in the context of inequalities lies in the conditions required for a solution to be valid: "or" requires at least one inequality to be true, while "and" requires all inequalities to be true simultaneously.
"Or" Inequalities
An "or" inequality presents two or more inequalities joined by the word "or." A solution to an "or" inequality must satisfy at least one of the inequalities. This means if a value makes one inequality true, it is a solution, even if it makes the other inequality false. It can also satisfy both.
- Condition: At least one inequality must be true.
- Solution Set: Includes all values that satisfy either inequality.
- Graphical Representation: The solution set is the union of the solution sets of the individual inequalities. The solutions are going in different directions (usually)
Example:
x < 2 or x > 5
A value like 1 satisfies x < 2, so it's a solution. A value like 6 satisfies x > 5, so it's also a solution. Values between 2 and 5 (inclusive) are not solutions.
"And" Inequalities
An "and" inequality presents two or more inequalities joined by the word "and." A solution to an "and" inequality must satisfy all of the inequalities simultaneously.
- Condition: All inequalities must be true.
- Solution Set: Includes only values that satisfy all inequalities.
- Graphical Representation: The solution set is the intersection of the solution sets of the individual inequalities. The solutions are usually in the middle and the two inequalities are going in opposite directions.
Example:
x > 2 and x < 5
Only values between 2 and 5 (exclusive) are solutions. A value like 1 does not satisfy x > 2. A value like 6 does not satisfy x < 5. Only values such as 3 or 4 satisfy both.
Summary Table
| Feature | "Or" Inequality | "And" Inequality |
|---|---|---|
| Connecting Word | or | and |
| Condition | At least one inequality must be true. | All inequalities must be true. |
| Solution Set | Union of individual solution sets. | Intersection of individual solution sets. |
| Satisfying | Satisfies either inequality | Satisfies both inequalities simultaneously |
