Expressing Real Number Sets and Compound Inequalities using Interval Notation page 6

Fundamentals of Interval Notation for Compound Inequalities

• Definition of Interval Notation: A system for describing subsets of the real number line using brackets and parentheses to indicate whether endpoints are included or excluded.

  • Square Brackets $[$ or $]$: Used when an endpoint is included in the set, corresponding to non-strict inequalities ($\le$ or $\ge$).

  • Parentheses $($ or $)$: Used when an endpoint is excluded from the set, corresponding to strict inequalities ($<$ or $>$), or when referencing infinity ($\infty$ or $-\infty$).

  • Infinity Symbols ($\infty$ and $-\infty$): Represent unbounded sets in the positive or negative direction and are always accompanied by parentheses.

• Expressing Union of Disjoint Intervals:

  • The word "or" between inequality statements signifies that a number can satisfy either condition, corresponding to the logical set operation of union.

  • The union symbol $\cup$ is placed between separate interval expressions to combine them into a single mathematical statement.

Worked Example: Expressing Real Numbers Less Than or Equal to $a$ or Greater Than or Equal to $b$

• Problem Statement:

  • Write the interval expressing all real numbers less than or equal to $-1$ or greater than or equal to $1$.

• Step-by-Step Mathematical Analysis:

  • Analyzing the First Component:

  • Inequality statement: All real numbers less than or equal to $-1$, written algebraically as $x \le -1$.

  • Boundary values: The interval extends from negative infinity $-\infty$ up to and including $-1$.

  • Interval expression for the first part: $(-\infty, -1]$.

  • Analyzing the Second Component:

  • Inequality statement: All real numbers greater than or equal to $1$, written algebraically as $x \ge 1$.

  • Boundary values: The interval starts at and includes $1$, extending to positive infinity $\infty$.

  • Interval expression for the second part: $[1, \infty)$.

  • Combining with the Union Operator:

  • Because the condition uses "or", the complete set is the union of both sub-intervals.

  • Final Interval Notation: $(-\infty, -1] \cup [1, \infty)$.

Practice Problems and Solutions

• Try It Problem 1:

  • Problem Statement: Express all real numbers less than $-2$ or greater than or equal to $3$ in interval notation.

  • Step-by-Step Solution:

  • First condition: Numbers less than $-2$ are expressed algebraically as $x < -2$. Since $-2$ is not included (strict inequality), use a parenthesis: $(-\infty, -2)$.

  • Second condition: Numbers greater than or equal to $3$ are expressed algebraically as $x \ge 3$. Since $3$ is included (non-strict inequality), use a square bracket: $[3, \infty)$.

  • Union of intervals: Combine both parts using the union symbol $\cup$.

  • Final Solution: $(-\infty, -2) \cup [3, \infty)$.

• Try It Problem 2 (Alternative Version):

  • Problem Statement: Express the set $-3 \le x \le 6$ using interval notation.

  • Step-by-Step Solution:

  • Inequality Analysis: The compound inequality $-3 \le x \le 6$ represents a single bounded interval containing all real numbers between $-3$ and $6$, inclusive.

  • Endpoint Evaluation:

    • Lower bound: $-3$ is included due to the $\le$ operator, requiring a left square bracket $[$.

    • Upper bound: $6$ is included due to the $\le$ operator, requiring a right square bracket $]$.

  • Final Solution: $[-3, 6]$.