Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 15

Chapter 2, Problem 15

In Exercises 15–26, use graphs to find each set. (- 3, 0) ∩ [- 1, 2]

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Identify the two sets given: the first set is the open interval $(-3, 0)$, which includes all numbers greater than $-3$ and less than $0$, but does not include the endpoints $-3$ and $0$.

The second set is the closed interval $[-1, 2]$, which includes all numbers from $-1$ to $2$, including the endpoints $-1$ and $2$.

To find the intersection $(-3, 0) \cap [-1, 2]$, determine the numbers that are common to both sets. This means finding the overlap between the intervals on the number line.

Since $(-3, 0)$ goes from just greater than $-3$ up to just less than $0$, and $[-1, 2]$ goes from $-1$ to $2$, the overlapping region will start at the larger of the two lower bounds and end at the smaller of the two upper bounds.

Write the intersection interval using the appropriate interval notation, considering whether the endpoints are included or excluded based on the original sets.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Interval Notation

Interval notation is a way to represent sets of real numbers between two endpoints. Parentheses () indicate that an endpoint is not included (open interval), while brackets [] mean the endpoint is included (closed interval). For example, (-3, 0) includes all numbers greater than -3 and less than 0, but not -3 or 0.

Intersection of Sets

The intersection of two sets consists of all elements that are common to both sets. When dealing with intervals, the intersection is the overlapping portion of the intervals. For example, the intersection of (-3, 0) and [-1, 2] includes all numbers that lie in both intervals.

Graphing Intervals on the Number Line

Graphing intervals on a number line helps visualize the sets and their relationships. Open intervals are shown with open circles, and closed intervals with filled circles at endpoints. By graphing both intervals, you can easily identify their intersection by observing the overlapping region.

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