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 interval \((-\infty, 5)\) represents all numbers less than 5, and the interval \([1, 8)\) represents all numbers from 1 to 8, including 1 but not including 8.
2. Step 2: Visualize the intervals on a number line. Draw a number line and shade the region for \((-\infty, 5)\) to the left of 5, and for \([1, 8)\), shade from 1 to just before 8.
3. Step 3: Identify the intersection. The intersection \((-\infty, 5)\) ∩ \([1, 8)\) is the set of numbers that are in both intervals.
4. Step 4: On the number line, find the overlapping region of the two shaded areas. This is the section where both intervals overlap.
5. Step 5: Write the intersection as an interval. The overlapping region on the number line represents the intersection of the two sets.

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, while [1, 8) includes 1 and all numbers up to, but not including, 8.

Intersection of Sets

The intersection of two sets is the set of elements that are common to both sets. In the context of intervals, this means finding the values that belong to both intervals simultaneously. For instance, to find the intersection of (-∞, 5) and [1, 8), we look for numbers that are both less than 5 and at least 1.

Graphing Intervals

Graphing intervals involves representing the range of numbers visually 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 easily identifying overlaps and intersections between different intervals.