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Chapter 8: Problem 52

Concept Check Express each set in simplest interval form. [-9,1]\(\cup(-\infty,-3)\)

Short Answer

Expert verified
(-鈭�,-3)鈭猍-9,1闭

Step by step solution

01

- Identify Interval Boundaries

Examine each interval in the union. The first interval is \([-9,1]\), meaning it includes all numbers from -9 to 1, inclusive. The second interval is \((-\infty,-3)\), meaning it includes all numbers less than -3 but not including -3.
02

- Analysis of Overlapping Intervals

Since \((-\infty,-3)\) includes all numbers from negative infinity up to, but not including -3, and \([-9,1]\) starts at -9 and goes to 1, there is an overlap between \([-9,-3)\). However, these intervals are disjoint or non-overlapping in their respective inclusion points.
03

- Combine Intervals in Simplified Form

Combine the disjoint intervals into their simplest interval notation form: \((-\infty,-3) \cup [-9,1]\). This ensures all elements of both sets are included.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Union of Sets
When working with intervals, it is crucial to understand the concept of the union of sets. The union of two sets contains all elements that are in either one of the sets or in both.
For example, with intervals, you represent the union using the \cup symbol. It means combining all values from both intervals.
Let's consider the intervals \([-9,1]\) and \((-\backslashinfty,-3)\). The union is written as \([-9,1] \cup (-\backslashinfty,-3)\). This includes all numbers from both intervals.
Note that if the intervals have an overlap, every value within the overlap is counted once.
Interval Boundaries
Interval boundaries are the endpoints of an interval. They define the start and end points of the interval.
We use different notations to include or exclude these boundaries:
  • Square brackets \[ \] mean the boundary is included (closed interval).
  • Parentheses \( \) mean the boundary is not included (open interval).

For instance, in the interval \([-9,1]\), both -9 and 1 are included. However, in the interval \((-\backslashinfty,-3)\), -3 is not included.
Understanding these notations is key in correctly interpreting and simplifying sets.
Disjoint Intervals
Disjoint intervals are intervals that do not overlap. They do not share any common elements.
In our example, the intervals \([-9,1]\) and \((-\backslashinfty,-3)\) are mostly disjoint. They overlap in the range \([-9,-3)\), meaning they share some common elements but differ at -3.
    To simplify, respect the boundary conditions when combining. Hence, \((-\backslashinfty,-3) \cup [-9,1]\) ensures all elements from both intervals are included correctly.

Understand this concept helps prevent errors in set operations and interval notations.

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Most popular questions from this chapter

Graph each linear inequality. $$ 4 x-5 y>20 $$

Graph each linear inequality. $$ x+3 y \geq-2 $$

Graph the solution set, and write it using interval notation. $$ 10\left(\frac{1}{5} x+2\right)<10\left(\frac{1}{5} x+1\right) $$

Describe the region that is the solution set of each system of inequalities. $$ \begin{array}{l} x<0 \\ y>0 \end{array} $$

Graph the solution set of each system of linear inequalities. $$ \begin{array}{r} x+y<4 \\ x-y>-4 \\ y>-1 \end{array} $$
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