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

1. ## 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.

2. ## 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?

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

Originally Posted by xxStrikeback
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*}

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

Originally Posted by xxStrikeback
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.

5. ## 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.

6. ## Re: Thanks. Now for trig. function formulas.

Originally Posted by xxStrikeback
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.

$\displaystyle \sin{\left(\frac{\pi}{2}\right)} = 1$, I suggest you research the unit circle.

7. ## 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.

8. ## 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.

9. ## 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 $\displaystyle (\sin{\theta} + \cos{\theta})^2$.

10. ## 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.

11. ## 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?