Question

interval notation and infinite sets
algebra 1
sets of numbers that comprise intervals along a number line are of particular interest in mathematics. we have seen how to represent these intervals using set builder notation. now we will introduce an alternative called interval notation. in this notation, are used for closed circles and ( ) are used for open circles and the number line is omitted. the interval $-3 < x \leq 2$ would be written as $(-3, 2$.
exercise #1: sets representing intervals are shown on the number lines below. represent each set using set builder notation and interval notation.
table with columns: graphed interval (number lines), set builder notation, interval notation. first row: number line from -10 to 10, closed at -6, open at 8, handwritten \$-6 \leq x < 8$\, $\\{x \mid -6 \leq x < 8\\}$, $-6, 8)$. subsequent rows: number lines with various closed/open circles and arrows.

Explanation:

Step1: Identify 2nd interval endpoints

Closed circles at -2 and 6.

Step2: 2nd set builder and interval

Set: {x | -2 ≤ x ≤ 6}, Interval: [-2, 6]

Step3: Identify 3rd interval endpoints

Open at -10, closed at 10.

Step4: 3rd set builder and interval

Set: {x | -10 < x ≤ 10}, Interval: (-10, 10]

Step5: Identify 4th interval endpoints

Closed at -4, open at 6.

Step6: 4th set builder and interval

Set: {x | -4 ≤ x < 6}, Interval: [-4, 6)

Step7: Identify 5th interval endpoints

Open at 0, open at 8.

Step8: 5th set builder and interval

Set: {x | 0 < x < 8}, Interval: (0, 8)

Step9: Identify 6th interval endpoints

Open at -4, closed at 8.

Step10: 6th set builder and interval

Set: {x | -4 < x ≤ 8}, Interval: (-4, 8]

Step11: Identify 7th interval endpoints

Closed at -6, closed at 2.

Step12: 7th set builder and interval

Set: {x | -6 ≤ x ≤ 2}, Interval: [-6, 2]

Step13: Identify 8th interval endpoints

Open at -4, closed at 2, open at 8 (union).

Step14: 8th set builder and interval

Set: {x | -4 < x ≤ 2 or x > 8}, Interval: (-4, 2] ∪ (8, ∞)

Answer:

2nd row: Set Builder Notation: {x | -2 ≤ x ≤ 6}, Interval Notation: [-2, 6]
3rd row: Set Builder Notation: {x | -10 < x ≤ 10}, Interval Notation: (-10, 10]
4th row: Set Builder Notation: {x | -4 ≤ x < 6}, Interval Notation: [-4, 6)
5th row: Set Builder Notation: {x | 0 < x < 8}, Interval Notation: (0, 8)
6th row: Set Builder Notation: {x | -4 < x ≤ 8}, Interval Notation: (-4, 8]
7th row: Set Builder Notation: {x | -6 ≤ x ≤ 2}, Interval Notation: [-6, 2]
8th row: Set Builder Notation: {x | -4 < x ≤ 2 or x > 8}, Interval Notation: (-4, 2] ∪ (8, ∞)