- Use the set-builder notation \(\{x \mid \dots\}\) to describe a set of elements satisfying a specific condition.
- Use interval notation \((a, b)\) or \([a, b]\) to represent continuous ranges, where parentheses indicate exclusivity and brackets indicate inclusivity.
- For given inequalities, translate to set-builder notation \(\{x \mid \text{inequality}\}\) and interval notation based on the bounds.
- For given interval or set-builder notations, reverse the process to find the corresponding inequality.
- The final table rows are populated as:
- 2: \(\{x \mid x < 6\}\), \((-\infty, 6)\)
- 3: \(x \le 3\), \((-\infty, 3]\)
- 4: \(\{x \mid x \le -3\}\), \((-\infty, -3]\)
- 5: \(x > 4\), \(\{x \mid x > 4\}\)
- 6: \(x \ge -2\), \([-2, \infty)\)
- 9: \(\{x \mid x < 2 \text{ or } x \ge 4\}\), \((-\infty, 2) \cup [4, \infty)\).
Explanation
- Problem Analysis and Definitions
To fill in the table, we use the standard definitions of inequalities, set-builder notation, and interval notation.
- An inequality describes a range of values for a variable (e.g., \(x < 6\)).
- Set-builder notation uses the form \(\{x \mid \text{condition on } x\}\), where the condition is the same as the inequality.
- Interval notation represents a range using parentheses for open bounds (exclusive) and brackets for closed bounds (inclusive), with \(\infty\) or \(-\infty\) for unbounded directions.
For each item:
- Item 1: 'r :' is given, which is a label, not an inequality. We will leave this blank as it is not specified.
- Item 2: \(x < 6\). Set notation is \(\{x \mid x < 6\}\). Interval notation is \((-\infty, 6)\).
- Item 3: The given information is contradictory (inequality \(x \ge 5\) vs set notation \(\{x \mid x \le 3\}\)). We will treat each row based on the most informative given field. For item 3, the set notation \(\{x \mid x \le 3\}\) leads to interval notation \((-\infty, 3]\).
- Item 4: \(x \le -3\). Set notation is \(\{x \mid x \le -3\}\). Interval notation is \((-\infty, -3]\).
- Item 5: Interval notation is \((4, \infty)\). Inequality is \(x > 4\). Set notation is \(\{x \mid x > 4\}\).
- Item 6: Set notation is \(\{x \mid x \ge -2\}\). Inequality is \(x \ge -2\). Interval notation is \([-2, \infty)\).
- Item 7 & 8: These are blank and have no provided info, so they will be left blank.
- Item 9: \(x < 2\) or \(x \ge 4\). Set notation is \(\{x \mid x < 2 \text{ or } x \ge 4\}\). Interval notation is \((-\infty, 2) \cup [4, \infty)\).
- Completing the Table Data
Based on the analysis, we complete the table as follows:
- Item 2: Inequality \(x < 6\), Set \(\{x \mid x < 6\}\), Interval \((-\infty, 6)\).
- Item 3: Given Set \(\{x \mid x \le 3\}\), Inequality \(x \le 3\), Interval \((-\infty, 3]\).
- Item 4: Inequality \(x \le -3\), Set \(\{x \mid x \le -3\}\), Interval \((-\infty, -3]\).
- Item 5: Interval \((4, \infty)\), Inequality \(x > 4\), Set \(\{x \mid x > 4\}\).
- Item 6: Set \(\{x \mid x \ge -2\}\), Inequality \(x \ge -2\), Interval \([-2, \infty)\).
- Item 9: Inequality \(x < 2 \text{ or } x \ge 4\), Set \(\{x \mid x < 2 \text{ or } x \ge 4\}\), Interval \((-\infty, 2) \cup [4, \infty)\).
Examples
Understanding these notations is fundamental for describing domains of functions or solutions to inequalities. For instance, if a store has a discount for purchases over \$50, the inequality \(x > 50\) defines the eligible range, which can be represented as the set \(\{x \mid x > 50\}\) or the interval \((50, \infty)\). Mastering these forms ensures clear communication in mathematics and real-world applications.