Overview
This lecture covers interval notation for expressing solution sets of inequalities, how to graph these on number lines, and the conversion between inequalities and interval notation.
Interval Notation Basics
- Interval notation expresses sets of numbers between two endpoints.
- Parentheses ( ) mean a boundary is not included; brackets [ ] mean it is included.
- Infinity (∞) always uses parentheses, since it cannot be included.
Graphing Intervals on Number Lines
- To graph (a, b), use open circles (parentheses) at a and b and shade in between.
- To graph [b, ∞), use a closed circle (bracket) at b and shade to the right.
- The entire set of real numbers is written as (−∞, ∞).
Translating Inequalities to Interval Notation
- x > 8 becomes (8, ∞) on a number line, shaded rightward from 8 with a parenthesis.
- x < 4 and x ≤ −2 (with "and") means take the overlap; solution is (−∞, −2] (bracket at −2).
- x > 4 or x ≤ −2 ("or") becomes (−∞, −2] ∪ (4, ∞).
- 0 ≤ x < 6 is written as [0, 6) and shaded between 0 (bracket) and 6 (parenthesis).
From Interval Notation to Graphs
- For (3, 9), bracket or parenthesis style matches the interval, and only the region between is shaded.
- (−∞, −3] is shaded all the way left with a bracket at −3.
Solving Inequalities and Writing the Interval
- Solve 4(x − 1) − 5 > 9x + 1 by distributing, combining like terms, and isolating x.
- After solving, result is x < −2, which in interval notation is (−∞, −2).
Key Terms & Definitions
- Interval Notation — A way to express sets of numbers between endpoints using parentheses and brackets.
- Bracket [ ] — Includes the endpoint in the interval.
- Parenthesis ( ) — Excludes the endpoint from the interval.
- Union ( ∪ ) — Combines two separate intervals.
- Real Numbers (ℝ) — All numbers on the number line, often written as (−∞, ∞).
Action Items / Next Steps
- Practice converting inequalities to interval notation and graphing them.
- Complete assigned homework exercises on interval notation (if specified by instructor).