IndietroSolving Linear Inequalities and Interval Notation
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Linear Inequalities and Interval Notation
Introduction to Linear Inequalities
Linear inequalities are mathematical statements that compare linear expressions using inequality symbols such as >, <, ≥, and ≤. The solution to a linear inequality is a set of values that make the inequality true, often represented on a number line and using interval notation.
Solving and Representing Linear Inequalities
Standard Notation: The inequality is written using symbols (e.g., x > 2).
Number Line Representation: Solutions are shown as arrows or shaded regions on a number line, indicating which values of x satisfy the inequality.
Interval Notation: Solutions are expressed as intervals, using parentheses ( ) for open endpoints and brackets [ ] for closed endpoints.
Examples and Interval Notation
Inequality (Standard Notation) | Number Line Representation | Interval Notation |
|---|---|---|
x > 2 | Arrow to the right from 2 (open circle) | (2, infty) |
x < 2 | Arrow to the left from 2 (open circle) | (-infty, 2) |
x ≥ 2 | Arrow to the right from 2 (closed circle) | [2, infty) |
x ≤ 2 | Arrow to the left from 2 (closed circle) | (-infty, 2] |
-2 < x < 2 | Arrow between -2 and 2 (open circles) | (-2, 2) |
-2 ≤ x ≤ 2 | Arrow between -2 and 2 (closed circles) | [-2, 2] |
Key Concepts and Definitions
Open Interval: means all values between a and b, not including endpoints.
Closed Interval: means all values between a and b, including endpoints.
Infinity: is always paired with a parenthesis, never a bracket, because infinity is not a specific number.
Compound Inequality: An inequality that combines two statements, such as .
Extra Notes
Intervals ending in : Use a parenthesis, e.g., .
Intervals starting with : Use a parenthesis, e.g., .
Compound inequalities: Represented as intervals, e.g., is .
Application Example
Example: Solve and write the solution in interval notation. Solution: The solution is all real numbers greater than or equal to 3. Interval Notation:
Additional info: Interval notation is a concise way to describe sets of numbers and is widely used in algebra to express solution sets for inequalities.
