A compound inequality is created by joining two or more inequalities together using "and" or "or." Here's how to make one:
Understanding Compound Inequalities
- "And" inequalities: Represent values that satisfy both inequalities simultaneously. The solution is the intersection of the two inequalities. They are often written as a single statement like a < x < b.
- "Or" inequalities: Represent values that satisfy at least one of the inequalities. The solution is the union of the two inequalities.
Steps to Create a Compound Inequality
-
Choose your inequalities: Select two (or more) simple inequalities. These can be any combination of:
>(greater than)<(less than)>=(greater than or equal to)<=(less than or equal to)
Example:
- Inequality 1:
x > 3 - Inequality 2:
x < 7
-
Choose your connective (and/or): Decide whether the solution needs to satisfy both inequalities ("and") or at least one of them ("or").
Example: Let's use "and."
-
Combine the inequalities: Write the two inequalities connected by "and" or "or."
Example:
x > 3 and x < 7 -
(Optional) Simplify (for "and" inequalities): If the "and" inequality can be simplified into a single statement, do so. This is possible when both inequalities involve the same variable and create a continuous interval.
Example:
x > 3 and x < 7can be simplified to3 < x < 7. This reads as "x is greater than 3 and less than 7."
Examples
Example 1: "And" Inequality
- Inequality 1:
y >= -2 - Inequality 2:
y <= 5 - Connective: "and"
- Compound Inequality:
y >= -2 and y <= 5 - Simplified:
-2 <= y <= 5
Example 2: "Or" Inequality
- Inequality 1:
a < 0 - Inequality 2:
a > 4 - Connective: "or"
- Compound Inequality:
a < 0 or a > 4(This cannot be simplified further.)
Example 3: "And" Inequality with No Solution
- Inequality 1:
z > 5 - Inequality 2:
z < 2 - Connective: "and"
- Compound Inequality:
z > 5 and z < 2(There is no solution, as no number can be both greater than 5 and less than 2.)
By combining two or more inequalities with "and" or "or," you create a compound inequality that represents a specific range or set of values for a variable.
