In Exercises 15–26, use graphs to find each set. [3, ∞) ⋃ (6, ∞)

검증된 단계별 안내

Understand the problem: You are asked to find the union of two intervals, \([3, \infty)\) and \((6, \infty)\), using graphs. The union means all values that belong to either interval or both.

Graph the first interval \([3, \infty)\) on a number line. This includes all numbers starting at 3 (including 3) and extending to positive infinity. Represent this with a solid dot at 3 and a ray extending to the right.

Graph the second interval \((6, \infty)\) on the same number line. This includes all numbers greater than 6 but not including 6. Represent this with an open dot at 6 and a ray extending to the right.

Identify the union of these two intervals on the graph. Since \([3, \infty)\) already includes all numbers from 3 onwards, and \((6, \infty)\) is a subset of that starting from 6, the union will cover all numbers starting at 3 and going to infinity.

Write the union interval notation based on the graph: it will be \([3, \infty)\) because this interval already contains all elements of both sets.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Interval Notation

Interval notation is a way to represent sets of real numbers using brackets and parentheses. Square brackets [ ] include the endpoint, while parentheses ( ) exclude it. For example, [3, ∞) includes all numbers from 3 to infinity, including 3.

Union of Sets

The union of two sets combines all elements from both sets without duplication. In interval notation, the union symbol (∪) merges intervals, so [3, ∞) ∪ (6, ∞) includes all numbers from 3 onwards, considering overlaps.

Graphing Intervals on the Number Line

Graphing intervals involves shading regions on a number line to represent sets of numbers. Closed dots indicate included endpoints, open dots indicate excluded endpoints, and shading shows all numbers within the interval, helping visualize unions and intersections.