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Math Help - Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

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    Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    Sorry for the first thread. Anyways, I am really struggling with these problems.

    I will post two for now, hopefully the work here will help me figure out the others.

    First, one division problem with several exponents.

    { (x^2 - 1)^1/2 - (x^2 - 1)^-1/2 (2 - x^2) } / (x^2 - 1) = 0

    The hint is be sure to simplify first. I have no idea how to start. Do I take the negative exponents to the bottom? Or cancel out? No clue.

    Second.

    | 1 - 4x | - 7 < or = - 2

    I'm not remembering the rules on abs. val. when there are variables in the bars, I know for |-2| it just becomes 2, |2| is 2, etc. But when there's variables in there, I'm not sure, so a link or explanation of that would really help.

    Thanks in advance, I hope this satisfies the rules.
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    Re: Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    Have a look here,

    http://www.mathhelpforum.com/math-he...tml#post676002

    For the second one you have

    | 1 - 4x | - 7 \leq - 2

    | 1 - 4x | - 5 \leq 0

    So define f(x) = | 1 - 4x | - 5 , where is this function less than zero?
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    Re: Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    Quote Originally Posted by xxStrikeback View Post
    Sorry for the first thread. Anyways, I am really struggling with these problems.

    I will post two for now, hopefully the work here will help me figure out the others.

    First, one division problem with several exponents.

    { (x^2 - 1)^1/2 - (x^2 - 1)^-1/2 (2 - x^2) } / (x^2 - 1) = 0

    The hint is be sure to simplify first. I have no idea how to start. Do I take the negative exponents to the bottom? Or cancel out? No clue.

    Second.

    | 1 - 4x | - 7 < or = - 2

    I'm not remembering the rules on abs. val. when there are variables in the bars, I know for |-2| it just becomes 2, |2| is 2, etc. But when there's variables in there, I'm not sure, so a link or explanation of that would really help.

    Thanks in advance, I hope this satisfies the rules.
    An alternative for the second is

    \displaystyle \begin{align*} |1 - 4x| - 7 &\leq -2 \\ |1 - 4x| &\leq 5 \\ -5 \leq 1 - 4x &\leq 5 \\ -6 \leq -4x &\leq 4 \\ \frac{3}{2} \geq x &\geq -1 \\ -1 \leq x &\leq \frac{3}{2}  \end{align*}
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    Re: Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    Quote Originally Posted by xxStrikeback View Post
    Sorry for the first thread. Anyways, I am really struggling with these problems.

    I will post two for now, hopefully the work here will help me figure out the others.

    First, one division problem with several exponents.

    { (x^2 - 1)^1/2 - (x^2 - 1)^-1/2 (2 - x^2) } / (x^2 - 1) = 0
    \frac{a^r- a^{-r}b}{a}= \frac{a^r}{a}- \frac{a^{-r}}{a}b= a^{r-1}- a^{-r-1}b

    | 1 - 4x | - 7 < or = - 2
    |1- 4x|\le 7- 2= 5
    First check where the two sides are equal.

    Then either 1- 4x= 5 or 1- 4x= -5.
    If 1- 4x= 5, -4x= 4, x= -1. If 1- 4x= -5, -4x= -6, x= 3/2.

    x= -5 and x= 3/2 divides the real line into three intervals: x\le -5, -5\le x\le 3/2, [tex]3\le x[/itex].
    x= -6 is in the first interval and if x= -6, |1- 4x|- 7= |1+ 24|- 7= 25- 7= 18> -2. x= -6 does NOT satisfy the inequality so no x< -5 does. Of course, x= -5 makes the two sides equal.

    If x= 0, which is between -5 and 3/2, then |1- 4x|- 7= |1- 0|- 7= -7< 2. x= 0 DOES satisfy the inequality so any x in -5\le x\le 3/2 does also.

    If x= 2, which is larger than 3/2, then |1- 4x|- 7= |1- 8|- 7= 7- 7= 0> -2. x= 2 does NOT satisfy the inequality so any x larger than 3/2 does not.

    The solution set is -5\le x\le 3/2.

    I'm not remembering the rules on abs. val. when there are variables in the bars, I know for |-2| it just becomes 2, |2| is 2, etc. But when there's variables in there, I'm not sure, so a link or explanation of that would really help.

    Thanks in advance, I hope this satisfies the rules.
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    Thanks. Now for trig. function formulas.

    Thanks for the help guys. I was able to work them out and get the right answers. Some of the stuff is starting to come back. Anyways, I know have some questions on this stuff.

    To be honest, not really sure where to go or what exactly to do. All I'm really getting is that [I figure] there is something to do with sin of pi/2 being 1 that is involved, but really not sure.

    Here.

    Algebra review Qs pt 2: divsion with several var./exp. &amp; abs. val. inequal.-7.png
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    Re: Thanks. Now for trig. function formulas.

    Quote Originally Posted by xxStrikeback View Post
    Thanks for the help guys. I was able to work them out and get the right answers. Some of the stuff is starting to come back. Anyways, I know have some questions on this stuff.

    To be honest, not really sure where to go or what exactly to do. All I'm really getting is that [I figure] there is something to do with sin of pi/2 being 1 that is involved, but really not sure.

    Here.

    Click image for larger version. 

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ID:	22150
    \displaystyle \sin{\left(\frac{\pi}{2}\right)} = 1, I suggest you research the unit circle.
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    Re: Thanks. Now for trig. function formulas.

    For example (b):
    \tan(x+\pi)=\tan(x)
    Write by using the subtraction formulas:
    \tan(x+\pi)=\frac{\sin(x+\pi)}{\cos(x+\pi)}=\frac{  \sin(x)\cdot \cos(\pi)+\sin(\pi)\cdot \cos(x)}{\cos(x)\cdot \cos(\pi)-\sin(\pi)\cdot \sin(x)}
    =\frac{-\sin(x)}{-\cos(x)}=\tan(x)}

    Try the others.
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    Re: Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    Thanks again for the help, I was able to get them and progress.

    Stumped again a couple problems later though.

    Really need to review trig stuff.

    Not sure how to start/do these.

    Algebra review Qs pt 2: divsion with several var./exp. &amp; abs. val. inequal.-untitled.png
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    Re: Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    1. Start by noting that \displaystyle \frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}. How can you use this to help you in the first problem?

    2. Start by using the double angle identity for \displaystyle \cos{2x} and the triple angle identity for \displaystyle \sin{3x}.

    3. Start by expanding \displaystyle (\sin{\theta} + \cos{\theta})^2.
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    Re: Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    Thanks, got it.

    Really appreciate all the help. Btw just so you guys know you are not doing all my homework haha, I do about 10-15 problems in between these.

    Here's the other two I don't know how to do. I completely forgot rules of trig so I don't know how to cancel stuff out or the trig formulas.
    Attached Thumbnails Attached Thumbnails Algebra review Qs pt 2: divsion with several var./exp. &amp; abs. val. inequal.-8.png  
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    Re: Algebra review Qs pt 2: divsion with several var./exp. & abs. val. inequal.

    For the first one consider \displaystyle \tan x = \frac{\sin x}{\cos x}

    \frac{\sin x +\tan x \cos x}{\tan x} =\frac{\sin x }{\tan x}+\frac{\tan x \cos x}{\tan x} = \cdots



    For the second make \sin 3x = \sin (2x+x) and apply the compound angle formula




    For the third one, expand the LHS, what do you get?





    Finally \displaystyle \cot 2x = \frac{-1}{\sqrt{3}}\implies \frac{1}{\tan x} = \frac{-1}{\sqrt{3}}\implies \tan x = -\sqrt{3}




    Maybe time you started a new thread?
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