Summer AP Calc Homework [Urgent]

[jbauman793]

I was given summer calc homework to help prepare for the year. The first problem is extremely confusing! I have never, ever come across a problem like this before in any of my math classes, and we were not given books to help us, so I'm stuck. Please help me? Thanks in advance!

1). If 2 < x < 6, which of the following statements about x are necessarily true and which are not necessarily true? Use NT for necessarily true and NNT for not necessarily true.

a). 0 < x < 4
b). 0 < x-2 < 4
c). 1< x/2 < 3
d). 1/6 < 1/x < 1/2
e). 1< 6/x < 3
f). |x-4| < 2
g). -6 < -x <2
h). -6 < -x < -2

That is the whole problem. I tried to figure out a. I thought maybe since 2 < x < 6, that x= 3, 4, or 5. Since a reads 0 < x < 4... that x would have to be 3...but the original its 3 4 and 5. So I put that its NNT. Now I'm pretty sure that I did that all wrong to begin with....but its all have have to start out with.

 

[Deleted member 4993]

I was given summer calc homework to help prepare for the year. The first problem is extremely confusing! I have never, ever come across a problem like this before in any of my math classes, and we were not given books to help us, so I'm stuck. Please help me? Thanks in advance!

1). If 2 < x < 6, which of the following statements about x are necessarily true and which are not necessarily true? Use NT for necessarily true and NNT for not necessarily true.

a). 0 < x < 4
b). 0 < x-2 < 4
c). 1< x/2 < 3
d). 1/6 < 1/x < 1/2
e). 1< 6/x < 3
f). |x-4| < 2
g). -6 < -x <2
h). -6 < -x < -2

That is the whole problem. I tried to figure out a. I thought maybe since 2 < x < 6, that x= 3, 4, or 5. Since a reads 0 < x < 4... that x would have to be 3...but the original its 3 4 and 5. So I put that its NNT. Now I'm pretty sure that I did that all wrong to begin with....but its all have have to start out with.

Plot 2<x<6 on a number line - if you don't know how to do that - for a quick review go to: http://www.purplemath.com/modules/ineqlin.htm

Then plot each of the possible answers given.

If the domain of given answer falls completly within the domain of 2<x<6 - then it is necessarily true.

If the domain of given answer falls completly outsidethe domain of 2<x<6 - then it is necessarily false.

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

 

[jbauman793]

So for the original number line...there would be an open circle on 2 and 6? and they'd be connected in the middle? Okay I get making the number line. I don't really have any work to show because I was only able to kind of understand letter a. For a... the domain would be 1 2 or 3? I'm also stuck on finding the domain of each set of integers (a-h). I'm not asking for the answers, I'm just asking for help to get me started so I can move on in the packet.

 

[jbauman793]

I think I may have figured it out. Can somebody tell me just so I'm compleletly sure?
a would be NNT Because all three numbers of the domain (123) are not COMPLELETLY in the domain of the original (345)
but b would be NT because if the original was subtracted by 2 then the domain would now be (123) so the domain of b would be complelety in it.
Is this true?

 

[Deleted member 4993]

I think I may have figured it out. Can somebody tell me just so I'm compleletly sure?
a would be NNT Because all three numbers of the domain (123) are not COMPLELETLY in the domain of the original (345)
but b would be NT because if the original was subtracted by 2 then the domain would now be (123) so the domain of b would be complelety in it.
Is this true?

Your logic is sort of correct - but why are you assuming'x' to be integer?

 

[mmm4444bot]

Subhotosh, are you saying that not necessarily false is necessarily true ?

< smirk >

 

[mmm4444bot]

b would be NT because if the original was subtracted by 2 then the domain would now be (123)

Yes, this conditional shift to the left works. But I think the other way around because I try to avoid changing givens.

Are you instructed to solve this exercise graphically using shifts? The exercise can be done algebraically.

0 < x - 2 < 4

This compound inequality statement tells us that the number x - 2 lies between 0 and 4. It does not explicitly tell us where the number x lies.

We can solve this compound inequality statement for x.

Add 2 throughout:

0 + 2 < x - 2 + 2 < 4 + 2

2 < x < 6

We see that (b) is NT because it's the same interval as the given.

For part (c), 1 < x/2 < 3

Multiply by 2 throughout:

1(2) < (x/2)(2) < 3(2)

2 < x < 6

We see that (c) is NT because it's the same interval as the given.

Eyeballing each of the remaining parts, I suspect that most of them are the same, too.

Hint: Make sure you know about the special rules (i.e., multiplying or dividing through by a negative number, changing direction with reciprocals, changing absolute value statements to inequalities).