Overview
This lesson covers how to write, interpret, and graph linear inequalities, as well as how to write inequalities based on graphs.
Understanding Inequalities
- An inequality compares two expressions using symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
- Common phrases indicating inequalities: "is fewer than" (<), "is more than" (>), "is at most" (≤), "is at least" (≥).
- Less than symbol (<) points left; greater than symbol (>) points right.
Writing Inequalities from Sentences
- Example: "A number w minus 3.5 is less than or equal to –2" is written as w – 3.5 ≤ –2.
- Example: "3 is less than a number n plus 5" is written as 3 < n + 5.
- Example: "0 is greater than or equal to twice a number x plus one" is written as 0 ≥ 2x + 1.
Checking Solutions to Inequalities
- Substitute the given value into the inequality.
- If the resulting statement is true, the value is a solution; if false, it is not.
- Example: For x + 8 < –3, x = –4 gives 4 < –3 (false, so not a solution).
- Example: For –4.5x > –21, x = –4 gives 18 > –21 (true, so it is a solution).
Graphing Inequalities on a Number Line
- Draw a number line with at least three values, centering on the key number.
- Use a closed circle (●) for "or equal to" (solution included); open circle (○) when not included.
- Draw an arrow left for "less than" (<, ≤) and right for "greater than" (>, ≥).
- Example: y ≤ –3 is graphed with a closed circle at –3, arrow left.
- Example: x > 2 is graphed with an open circle at 2, arrow right.
- Example: x > 0 is graphed with an open circle at 0, arrow right.
Writing Inequalities from Graphs
- Closed circle and arrow right: variable ≥ number.
- Open circle and arrow left: variable < number.
- Example: Closed circle at 48, arrow right is h ≥ 48.
- Example: Open circle at 52, arrow left is h < 52.
Key Terms & Definitions
- Inequality — A mathematical sentence comparing two expressions using <, >, ≤, or ≥.
- Solution — A value that makes the inequality true.
- Closed circle (●) — Indicates the endpoint is included (≤ or ≥) on the graph.
- Open circle (○) — Indicates the endpoint is not included (< or >) on the graph.
Action Items / Next Steps
- Practice writing and graphing inequalities from word problems and graphs.
- Complete assigned exercises on graphing and identifying solutions to inequalities.