1.2.5 Solution Sets and Interval Notation¶

Express inequality solutions using interval notation, set-builder notation, and identify empty sets or all real numbers.

定义¶

A solution set is the collection of all values that satisfy a given inequality or equation. Solution sets can be expressed in multiple notations:

Interval Notation: Uses parentheses and brackets to denote ranges of numbers. Parentheses ( ) indicate that an endpoint is NOT included (open interval), while brackets [ ] indicate that an endpoint IS included (closed interval). For example, the interval \([a, b)\) includes all real numbers \(x\) such that \(a \leq x < b\).
Set-Builder Notation: Expresses the solution set using the form \(\{x \mid \text{condition}\}\), which reads as "the set of all \(x\) such that the condition is true." For example, \(\{x \mid x > 3\}\) represents all real numbers greater than 3.
Special Cases:
Empty Set: Denoted as \(\emptyset\) or \(\{\}\), represents when no values satisfy the inequality (e.g., \(x > 5\) AND \(x < 2\)).
All Real Numbers: Denoted as \((-\infty, \infty)\) or \(\mathbb{R}\), represents when all real numbers satisfy the inequality (e.g., \(x > -\infty\) AND \(x < \infty\)).
Infinity: The symbols \(\infty\) and \(-\infty\) always use parentheses (never brackets) because infinity is not a real number and cannot be "reached."
Union of Intervals: When a solution set consists of multiple disjoint intervals, they are connected using the union symbol \(\cup\). For example, \((-\infty, -2) \cup (3, \infty)\) represents all numbers less than -2 or greater than 3.

\([a, b] = \{x \mid a \leq x \leq b\}\) (closed interval, both endpoints included)
\((a, b) = \{x \mid a < x < b\}\) (open interval, neither endpoint included)
\([a, b) = \{x \mid a \leq x < b\}\) (half-open interval, left endpoint included)
\((a, b] = \{x \mid a < x \leq b\}\) (half-open interval, right endpoint included)
\((-\infty, b] = \{x \mid x \leq b\}\) and \([a, \infty) = \{x \mid x \geq a\}\) (unbounded intervals)

⚠️ ["Using brackets [ ] with infinity symbols: Students often write \([3, \infty)\) when they mean \((3, \infty)\), or use brackets instead of parentheses with infinity. Remember: infinity ALWAYS requires parentheses because it is not a real number.", "Confusing the direction of inequality symbols with interval notation: When converting \(x < 5\) to interval notation, students may write \([5, \infty)\) instead of \((-\infty, 5)\). The interval should extend in the direction of the inequality.", "Incorrectly combining disjoint intervals without the union symbol: When a solution consists of separate intervals like \(x < -1\) OR \(x > 3\), students may write \((-1, 3)\) instead of \((-\infty, -1) \cup (3, \infty)\), which represents the opposite solution set.", "Misinterpreting endpoints when translating between notations: Students may confuse whether an endpoint is included or excluded, writing \((a, b]\) when the condition requires \([a, b)\), or vice versa. Always check: does the inequality use \(<\) or \(\leq\)?"]