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Interval Notation and Graphing

Sep 3, 2025

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).