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The solution to an inequality is [tex](-\infty, 6.5][/tex]. Is 6.5 a solution to the inequality? Explain your answer.

A. The solution includes numbers from negative infinity to 6.5. Because a bracket is used, the solution does not include 6.5.
B. The solution includes numbers from 6.5 to infinity. Because a parenthesis is used, the solution does not include 6.5.
C. The solution is the point [tex](-\infty, 6.5)[/tex]. Because a bracket is used, 6.5 is a solution to the inequality.
D. The solution includes numbers from negative infinity to 6.5. Because a bracket is used, the solution includes 6.5.

Sagot :

GinnyAnsweridn
Let's analyze the solution step-by-step:

1. The solution to the inequality is given by the interval [tex]\((-\infty, 6.5]\)[/tex].
2. In interval notation, a round bracket ( or ) means that the endpoint is not included in the interval, while a square bracket [ or ] means that the endpoint is included in the interval.
3. Here, we have the interval [tex]\((-\infty, 6.5]\)[/tex], where negative infinity is represented by a round bracket, meaning it is not an endpoint, so it is not included. On the other hand, 6.5 is represented by a square bracket ], meaning it is included in the interval.
4. Therefore, the solution includes all numbers from negative infinity up to and including 6.5.

With this understanding, we can conclude that 6.5 is indeed part of the solution to the inequality because the square bracket indicates that 6.5 is included.

Now, let's review the provided options one by one:

1. "The solution includes numbers from negative infinity to 6.5 . Because a bracket is used, the solution does not include 6.5."
- This statement is incorrect because a bracket indicates that the endpoint (6.5) is included.

2. "The solution includes numbers from 6.5 to infinity. Because a parenthesis is used, the solution does not include 6.5."
- This statement is incorrect and irrelevant, as the interval [tex]\((-\infty, 6.5]\)[/tex] does not describe a range from 6.5 to infinity.

3. "The solution is the point [tex]\((-\infty, 6.5) \)[/tex]. Because a bracket is used, 6.5 is a solution to the inequality."
- This statement is incorrect. It incorrectly describes the interval with a round bracket for 6.5. Moreover, [tex]\((-\infty, 6.5)\)[/tex] with parentheses would mean 6.5 is not included.

4. "The solution includes numbers from negative infinity to 6.5 . Because a bracket is used, the solution includes 6.5."
- This statement is correct and matches our understanding of interval notation. The bracket indicates that 6.5 is included in the solution.

Therefore, the correct statement is: "The solution includes numbers from negative infinity to 6.5 . Because a bracket is used, the solution includes 6.5."
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