Understanding Inequalities on a Number Line
Graphing inequalities on a number line is a fundamental skill in algebra. It provides a visual representation of all values that satisfy an inequality, making it easier to understand solution sets and communicate mathematical relationships.
Types of Inequalities
Strict Inequality (>, <)
Uses an open circle at the boundary point to indicate it is NOT included.
Non-strict Inequality (≥, ≤)
Uses a closed (filled) circle at the boundary point to indicate it IS included.
Compound Inequality
Combines two inequalities to define a bounded interval on the number line.
Interval Notation
Parentheses ( ) for open endpoints, brackets [ ] for closed endpoints.
How to Graph Inequalities
- Identify the boundary point(s) from the inequality.
- Determine whether each endpoint is open or closed based on the inequality symbol.
- Draw the number line and mark the endpoint(s).
- Shade the region representing all solutions.
- Write the solution in interval notation.
Open vs Closed Dots
An open dot (hollow circle) means the endpoint is NOT part of the solution. This occurs with strict inequalities (< and >). A closed dot (filled circle) means the endpoint IS part of the solution. This occurs with non-strict inequalities (≤ and ≥).
Interval Notation Conventions
Interval notation uses brackets and parentheses to express solution sets concisely. Use parentheses for open endpoints and infinity, and brackets for closed endpoints. Examples: (3, +infinity), (-infinity, -2], [-2, 5), [0, 10].
Applications
- Solving algebraic equations with inequality constraints
- Defining domain and range of functions
- Optimization problems in economics and engineering
- Statistics and probability (confidence intervals)
- Real-world constraints (speed limits, temperature ranges)