Question Find the intersection and union of the intervals and , and express the results in interval notation.
Studdy Solution
STEP 1
1. The notation and represent intervals on the real number line, where the parentheses indicate that the endpoints are not included in the interval.
2. The symbol represents the intersection of two sets, which is the set of elements that are common to both sets.
3. The symbol represents the union of two sets, which is the set of all elements that are in either set or in both.
4. Interval notation is used to express the solution, where the smallest number is written first, followed by a comma, and then the largest number. Parentheses or brackets are used to indicate whether the endpoints are included (brackets) or not included (parentheses).
STEP 2
1. Determine the intersection of the two intervals.
2. Determine the union of the two intervals.
High_Level_Step: 1
Determine the intersection of the two intervals .
STEP 3
The intersection of two intervals is the set of all points that are in both intervals. We look for the common elements between and .
STEP 4
The interval includes all numbers between -5 and 1, but not including -5 and 1 themselves.
STEP 5
The interval includes all numbers between -6 and -1, but not including -6 and -1 themselves.
STEP 6
The common elements between the two intervals are those that are greater than -5 and also less than -1.
STEP 7
Therefore, the intersection of and is the interval , since these are the numbers that are in both intervals.
Determine the union of the two intervals .
STEP 8
The union of two intervals is the set of all points that are in either interval or in both.
STEP 9
The interval includes all numbers between -5 and 1, but not including -5 and 1 themselves.
STEP 10
The interval includes all numbers between -6 and -1, but not including -6 and -1 themselves.
STEP 11
The union of the two intervals includes all numbers that are in either interval, which means we take the smallest number from both intervals and the largest number from both intervals to form the union.
STEP 12
The smallest number from both intervals is -6 and the largest number is 1.
STEP 13
Therefore, the union of and is the interval , since these are all the numbers that are in either interval.
The solutions are:
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