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Thread: Summer AP Calculus Homework [Urgent]

  1. #1
    Aug 2010

    Exclamation Summer AP Calculus Homework [Urgent]

    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.

    I understand that I have to draw a number line and find the domain...but I forget how to do both for (a-h). I don't believe I've ever had to do this!
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  2. #2
    Member mfetch22's Avatar
    Feb 2010
    Columbus, Ohio, USA
    I'll do the first one for you and explain why I got the answer I got. So, we are given the following:

    $\displaystyle 2 < x < 6$

    And for the first question we are given:

    $\displaystyle 0 < x < 4$

    This is not neccesarily true. Heres why; consider the following property of numbers $\displaystyle a, \; b, \; \mathrm{and} \; c$ which are greater then zero.


    $\displaystyle a<b$


    $\displaystyle a+c < b+c$

    Also, if:

    $\displaystyle a<b$


    $\displaystyle ac<bc$

    All the above holds given that:

    $\displaystyle 0<a \;\;\;\;\; 0<b \;\;\;\;\; 0<c \;\;\;\;\;$
    Now, with the above, remember that the problem started with:

    $\displaystyle 2<x<6$

    If problem (a) had said:

    $\displaystyle 0 < x - 2 < 4$

    then it would be neccesarily true, since we can start with:

    $\displaystyle 2 < x < 6$

    then get to it by:

    $\displaystyle 2 -2 < x -2 < 6 - 2$

    $\displaystyle 0 < x-2 < 4$

    But question (a) only said:

    $\displaystyle 0<x<4$

    Does that make sense? Now, using the properties of inequalities of positive numbers, see if you can't figure out the other questions. If your still stuck, post here and I'll give you more assitance
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  3. #3
    Senior Member
    Jan 2010
    If the above reply is helpful you can disregard this post, but I wanted give a simpler response in case it is more in line with what you wanted.

    First, the problem tells you that we are considering values of x that range between 2 and 6 not inclusive. So, that means that x could be 3, 4, 5, 2.1, 3.33333, 5.92341, or any other number between 2 and 6; however it cannot be exactly 2 or 6.

    Now let's look at the first one: (a) 0 < x < 4. We know that x can range between 2 and 6. Are there any values in this range that would NOT satisfy 0 < x < 4? In other words, are there any values of x between 2 and 6 that is not between 0 and 4? You can probably see that the answer is YES, because x=5 satisfies the first inequality but not part (a). Therefore you should respond NOT NECESSARILY TRUE.

    Your original idea was correct except that you should not limit yourself to only whole numbers!

    You should basically simplify each inequality given after that and then do the same comparison. It is really testing you on if you can simplify each inequality properly.
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