A combining inequality, also known as a compound inequality, is a mathematical statement that combines two or more inequalities using "and" or "or". Based on the reference provided, a clear example of a combining inequality is 1 < x < 3.
Understanding Combining Inequalities
Combining inequalities are fundamental in mathematics for expressing ranges of values. They help in describing conditions that must be met for a variable.
How They Work
A compound inequality joins individual inequalities. The two common joiners are "and" and "or".
- "And" Inequalities: These mean the variable must satisfy *both* conditions. For example, 1 < x < 3 means *x > 1* and *x < 3*.
- "Or" Inequalities: These mean the variable must satisfy *at least one* of the conditions. For example, *x < 1 or x > 3* means *x* is less than 1, or *x* is greater than 3.
Example: 1 < x < 3
The example, 1 < x < 3, is a compound inequality using "and". This is equivalent to saying:
x > 1 *and* x < 3
This means any value of 'x' that satisfies this compound inequality must be greater than 1 *and* less than 3. So, numbers such as 1.5, 2, and 2.9 all work.
Other Examples
- -2 ≤ y < 5 (y is greater than or equal to -2 AND y is less than 5)
- z < 0 or z > 10 (z is less than 0 OR z is greater than 10)
Table of Examples
| Compound Inequality | Meaning |
|---|---|
| 1 < x < 3 | x is greater than 1 AND x is less than 3 |
| -2 ≤ y < 5 | y is greater than or equal to -2 AND y is less than 5 |
| z < 0 or z > 10 | z is less than 0 OR z is greater than 10 |
In summary, combining inequalities are a useful way to describe a range or combination of ranges for a variable. The example 1 < x < 3 perfectly illustrates how two simple inequalities are combined.
