Quick Interval Notation Question

lolily

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Jan 28, 2019
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Hey! So if anyone has a spare minute, I just need to figure out how to express an overall answer to an equality problem in interval notation. I was asked to solve 6t≤7t-8 or 2−2t>20 seperately. I got t≥8 ( [8,∞ ) in interval notation) and t <-9 ( or (-∞ ,-9) ). Now that I have those answers, how do I use those answers to solve the overall problem and express that in interval notation? Sorry if this seems like a pretty easy question I've fallen behind because of sickness and have found it difficult to find anything else about this specifically on the internet. Thanks for the help in advance :) If I were to guess at the problem I would think that the answer was something like (-9,8] but I'm 99% sure that would be wrong.
 

Dr.Peterson

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Hey! So if anyone has a spare minute, I just need to figure out how to express an overall answer to an equality problem in interval notation. I was asked to solve 6t≤7t-8 or 2−2t>20 seperately. I got t≥8 ( [8,∞ ) in interval notation) and t <-9 ( or (-∞ ,-9) ). Now that I have those answers, how do I use those answers to solve the overall problem and express that in interval notation? Sorry if this seems like a pretty easy question I've fallen behind because of sickness and have found it difficult to find anything else about this specifically on the internet. Thanks for the help in advance :) If I were to guess at the problem I would think that the answer was something like (-9,8] but I'm 99% sure that would be wrong.
Since the problem was 6t ≤ 7t - 8 or 2 − 2t > 20, the solution is t ≥ 8 or t < -9. That is, any number that is either at least 8, or less than -9, is in the solution.

Your answer, (-9, 8], is the set of all values of t such that t 8 and t > -9. That is not what you want.

Try graphing your two inequalities. You'll see that they don't intersect; they cover two separate parts of the number line. But you don't want the intersection (where both are true). You want the set of all numbers for which at least one of them is true.

The set of all elements that are in set A or in set B is the union; so we use that for the answer: (-∞, -9) U [8, ∞). I'm guessing that you missed this lesson (and didn't see it in your textbook). Not all solution sets can be written as a single interval.
 

JeffM

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Hey! So if anyone has a spare minute, I just need to figure out how to express an overall answer to an equality problem in interval notation. I was asked to solve 6t≤7t-8 or 2−2t>20 seperately. I got t≥8 ( [8,∞ ) in interval notation) and t <-9 ( or (-∞ ,-9) ). Now that I have those answers, how do I use those answers to solve the overall problem and express that in interval notation? Sorry if this seems like a pretty easy question I've fallen behind because of sickness and have found it difficult to find anything else about this specifically on the internet. Thanks for the help in advance :) If I were to guess at the problem I would think that the answer was something like (-9,8] but I'm 99% sure that would be wrong.
We would need to see the exact wording of the problem. If the problem asks for the interval or intervals that satisfy both inequalities, you get one answer. If the problem asks for the interval or intervals that satisfy at least one of the equalities, you get a different answer.

This is why the guidelines ask for the complete and exact wording of the problem. Otherwise we must guess.

Please look at this summary of the guidelines.

https://www.freemathhelp.com/forum/threads/109845-Guidelines-Summary?p=422890&viewfull=1#post422890
 

lolily

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We would need to see the exact wording of the problem. If the problem asks for the interval or intervals that satisfy both inequalities, you get one answer. If the problem asks for the interval or intervals that satisfy at least one of the equalities, you get a different answer.

This is why the guidelines ask for the complete and exact wording of the problem. Otherwise we must guess.

Please look at this summary of the guidelines.

https://www.freemathhelp.com/forum/threads/109845-Guidelines-Summary?p=422890&viewfull=1#post422890

I did word the question exactly as my book did. I've read the guidelines before, thanks. The question is asking for one interval that satisfies both inequalities, just to clear things up.
 

lolily

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Since the problem was 6t ≤ 7t - 8 or 2 − 2t > 20, the solution is t ≥ 8 or t < -9. That is, any number that is either at least 8, or less than -9, is in the solution.

Your answer, (-9, 8], is the set of all values of t such that t 8 and t > -9. That is not what you want.

Try graphing your two inequalities. You'll see that they don't intersect; they cover two separate parts of the number line. But you don't want the intersection (where both are true). You want the set of all numbers for which at least one of them is true.

The set of all elements that are in set A or in set B is the union; so we use that for the answer: (-∞, -9) U [8, ∞). I'm guessing that you missed this lesson (and didn't see it in your textbook). Not all solution sets can be written as a single interval.

Thank you, this is what I was looking for and I understand the question much better now.
 

Dr.Peterson

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Nov 12, 2017
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... I was asked to solve 6t≤7t-8 or 2−2t>20 seperately. ...
I did word the question exactly as my book did. I've read the guidelines before, thanks. The question is asking for one interval that satisfies both inequalities, just to clear things up.
Let me explain why it is correct to say that you did not quote the problem exactly, and why it matters.

You gave only an "indirect quotation", making it a clause within your own sentence, rather than standing alone, suggesting that you may have paraphrased. In fact, you added the word "separately", which changes the meaning, implying that they want two separate solutions. Possibly you meant that in the instructions you were told to solve each part separately, and then combine them to make one answer. But what you wrote opened up the possibility that the problem did not say, literally, "Solve 6t≤7t-8 or 2−2t>20", as I assumed. That is what we mean by quoting exactly; we like to see the exact words used, and nothing else, to be sure we are interpreting it correctly, and don't have to make assumptions.

But now you say "one interval that satisfies both inequalities", which confuses things even more! If the question was as I assumed, then they would not want values that satisfy both inequalities; that would be the intersection of the solution sets, not the union. Also, as I explained, the solution is not a single interval!

Wording is very important here; students often confuse "and" and "or" in this way. I am again guessing that what you meant to say is that the question asks for a single solution for the compound inequality (that is, for the combination using "or"). Quoting word for word saves us the trouble of wondering if we are reading too much or too little into what you write.

I am emphasizing this not because you have done something terrible, but because others reading this may benefit from having the issue stated clearly.
 
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