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Chapter 2: Problem 52

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

Short Answer

Expert verified
(-鈭�,-3) 鈭� [-9,1]

Step by step solution

01

Identify the Intervals

The given sets are two intervals: 1. [-9,1] 2. (-fty,-3)
02

Determine if the Intervals Overlap

Check if there is any overlap between the two intervals. Since [-9,1] ends at 1 and (-fty,-3) extends all the way to negative infinity up to -3, these intervals do not overlap.
03

Combine the Intervals

Since there is no overlap, the union of these two intervals will be the intervals themselves listed together. Therefore, the combined interval in simplest form will be (-fty,-3) 鈭� [-9,1]

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Key Concepts

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

union of intervals
The union of intervals is a way to combine two or more intervals into a single set that covers all values from the given intervals. When taking the union of intervals, we are looking for all values that belong to at least one of the intervals.

For example, if we have intervals [-9,1] and (-fty,-3), the union represents all numbers that are either within [-9,1] or within (-fty,-3).

In interval notation, the union is denoted by the symbol (鈭�).

Here鈥檚 another example to illustrate: If you have the intervals [2,5] and [7,9], their union would be written as [2,5] 鈭� [7,9]. This means any number that is in the range [2,5] or in the range [7,9] is included in the union.
overlapping intervals
Overlapping intervals occur when two intervals share some common elements. In other words, the intervals have some numbers in common. Checking for overlapping intervals helps in simplifying the overall notation by merging parts that overlap.

For example, if we consider the intervals [1,5] and [3,7], they overlap because they both contain the numbers 3, 4, and 5. To represent the overlapping part, we combine the intervals to form the single interval [1,7].

To determine if intervals overlap, examine their endpoints:
  • The intervals [1,5] and [3,7] overlap because 5 > 3 and 1 < 7.
  • The intervals [-9,1] and (-fty,-3), as in the given exercise, do not overlap because their ranges are separate.
Understanding whether intervals overlap is crucial for writing them in the simplest form, as we would not list redundant or overlapping ranges separately.
simplifying interval notation
Simplifying interval notation involves combining intervals into their most reduced form, ensuring that there are no unnecessary overlaps or splits. This process helps make expressions easier to understand and use.

To simplify interval notation, follow these steps:
  • Identify each interval.
  • Check for any overlaps between the intervals.
  • If overlaps exist, merge them into a single interval.
  • If no overlaps exist, list the intervals distinctly using the union (鈭�) symbol.
Let鈥檚 revisit the exercise:

We have the intervals [-9,1] and (-fty,-3).
The endpoints of [-9,1] are -9 and 1, and the endpoints of (-fty,-3) extend to -3 without touching it.

Since there are no overlapping values, we directly combine the intervals using the union symbol:

(-fty,-3) 鈭� [-9,1].
This union notation shows that every number in either of those ranges is included in the final simplified set.

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