In Exercises 15–26, use graphs to find each set. (- ∞, 5) ⋃ [1, 8)

Verified step by step guidance

1. Step 1: Understand the notation. The expression \((-\infty, 5) \cup [1, 8)\) represents the union of two intervals on the number line.
2. Step 2: Identify the intervals. \((-\infty, 5)\) is an open interval, meaning it includes all numbers less than 5 but not 5 itself. \([1, 8)\) is a half-open interval, meaning it includes all numbers from 1 to 8, including 1 but not 8.
3. Step 3: Graph the intervals on a number line. For \((-\infty, 5)\), draw a line extending left from 5 with an open circle at 5. For \([1, 8)\), draw a line from 1 to 8 with a closed circle at 1 and an open circle at 8.
4. Step 4: Combine the intervals. The union \(\cup\) means you take all numbers that are in either interval. On the number line, this means shading all parts covered by either interval.
5. Step 5: Interpret the graph. The combined graph shows the set of all numbers less than 5, and also the numbers from 1 to 8, excluding 8. This is the set \((-\infty, 5) \cup [1, 8)\).

Key Concepts

Intervals

Intervals are a way to describe a range of numbers on the real number line. They can be open, closed, or half-open, depending on whether the endpoints are included. For example, the interval (-∞, 5) includes all numbers less than 5 but does not include 5 itself, while [1, 8) includes 1 and all numbers up to, but not including, 8.

Union of Sets

The union of sets combines all elements from the involved sets without duplication. In this case, the union of the intervals (-∞, 5) and [1, 8) means we take all numbers from both intervals, resulting in a continuous range that includes all numbers less than 5 and all numbers from 1 to just below 8.

Graphing Intervals

Graphing intervals involves visually representing the ranges of numbers on a number line. Open intervals are shown with parentheses (not including endpoints), while closed intervals use brackets (including endpoints). This visual representation helps in understanding the union of intervals by clearly showing which numbers are included in the final set.