Questions Regarding Solve, Simplify, Domain, Absolute Value

B

Blue Leader

New member
Joined
Feb 5, 2010
Messages
8
Greetings,

I'm back again with some more questions... Joy. :)

Once again I'm struggling with some various problems. There are a few of them so my apologies. I'm just rather stuck and not sure where to go with them.
I have been hitting the tutor, but I still am having some questions...

Simplify:
These ones I'm not exactly sure how to type, so please bare with me... I'll try typing these out and hopefully it'll make sense...
These are fractions within a fraction, which are really throwing me off. As if fractions weren't confusing enough to me as it is...

1.
\(\displaystyle \frac{x^2-x-12}{x^2-2x-15}\)
_______________
\(\displaystyle \frac{x^2+8x+12}{x^2-5x-14}\)

2.
\(\displaystyle \frac{1}{x-2} + \frac{3}{x-1}\)
_______________
\(\displaystyle \frac{2}{x-1} + \frac{5}{x-2}\)
For these, if I remember right I have to find a common denominator and multiply the problem by that... or something like that. Though I'm not sure how to find the common denominator for factions like these, or what to do with it from there...
And than I've got the whole fraction-within-a-fraction thing to worry about. Completely, utterly lost.

Solve:
3. \(\displaystyle y - \frac{14}{y} = 5\)

4. \(\displaystyle \frac{4}{a-7} = \frac{-2a}{a+3}\)
Again, for these I'd have to find a common denominator, correct? And than multiply everything by it? So number 3 the denominator would by Y and for number 4 the denominator would be... A? Maybe I'm no where close... :?

Simplify, use absolute value notation when necessary. If it cannot be simplified, state this:
5. \(\displaystyle \sqrt{9x^2y } \sqrt{3x^5y^2}\)

6. \(\displaystyle ^3\sqrt{ \frac{8m^7}{n^2} } ^3\sqrt{ \frac{n^9}{m^2} }\)
Just to clarify, those 3's in front of the square root are supposed to be cubes, I believe, not just the number 3.
The whole square root thing I'm having a problem with. I can find the square root of simple numbers, but toss in powers and multiple numbers and I start getting completely lost.


Determine the domain for each of the following:
7. \(\displaystyle g(x) = \sqrt {5x+2 }\)

8. \(\displaystyle d(x) = -\sqrt {7x-5 }\)
These ones don't look too complicated... and yet I have no clue where to go with them. Especially the "domain" thing... What's a "domain"? I don't remember going over that in class yet...
And for number 8, I didn't think you could get the square root of a negative number? So would that simply be marked "no solution"?

I know that this is a lot of problems that I've posted, but by no means do I expect help with all of them. I only thought I'd post what I am stuck on and perhaps I could get help for what people are willing to help me with.
But I am sorry for posting so many. Believe me, I am trying my best with these, and I do have over half of the worksheet finished for my college math class, but I'm still struggling with the ones I posted. I do feel bad for posting so many... I feel like I'm asking a lot.
Like I said, I have been going to a tutor, but unfortunately it's a public tutor and there's always a large crowd. After a while I tend to get tired of waiting for forty-five minutes just to have one question quickly answered.

Err, sorry for my rambling. But any help with any of the above problems would be appreciated.

Thank you for your time.
 
Covering all of these topics in a classroom takes several weeks. I don't have time to type up several weeks' worth of lessons.

Right now, I have time to comment on exercises (1), (3), and (4).

When one fraction is divided by another fraction, we call the "overall" fraction a compound fraction.

There is a simple rule, for dealing with compound fractions.

a/b divided by c/d is the same as a/b multiplied by d/c.

In other words, we don't divide by a fraction; we multiply by its reciprocal, instead.

In exercise (1), start by factoring each of polynomials.

There is a common factor to cancel, in the ratio of the two polynomials in the numerator of the compound fraction.

There is also a common factor to cancel, in the ratio of the two other polynomials in the denominator of the compound fraction.

After you cancel these common factors, you'll have the following.

Code: 
    x - 4
    -----
    x - 5
-------------
    x + 6
    -----
    x - 7

(I did this in my head, so please verify that my work is correct.)

Now, use the rule that tell us to multiply the ratio (x - 4)/(x - 5) by the reciprocal of the ratio (x + 6)/(x - 7).

Unless you've been told otherwise, you can leave your answer in factored form. 8-)

In exercise (3), multiply both sides of the equation by y, and solve the resulting quadratic equation.

In exercise (4), cross-multiply, and solve the resulting quadratic equation.
 
Blue Leader said:
Greetings,

I'm back again with some more questions... Joy. :)

Once again I'm struggling with some various problems. There are a few of them so my apologies. I'm just rather stuck and not sure where to go with them.
I have been hitting the tutor, but I still am having some questions...

Simplify:
These ones I'm not exactly sure how to type, so please bare with me... I'll try typing these out and hopefully it'll make sense...
These are fractions within a fraction, which are really throwing me off. As if fractions weren't confusing enough to me as it is...

1.
\(\displaystyle \frac{x^2-x-12}{x^2-2x-15}\)
_______________
\(\displaystyle \frac{x^2+8x+12}{x^2-5x-14}\)

2.
\(\displaystyle \frac{1}{x-2} + \frac{3}{x-1}\)
_______________
\(\displaystyle \frac{2}{x-1} + \frac{5}{x-2}\)
For these, if I remember right I have to find a common denominator and multiply the problem by that... or something like that. Though I'm not sure how to find the common denominator for factions like these, or what to do with it from there...
And than I've got the whole fraction-within-a-fraction thing to worry about. Completely, utterly lost.

Solve:
3. \(\displaystyle y - \frac{14}{y} = 5\)

4. \(\displaystyle \frac{4}{a-7} = \frac{-2a}{a+3}\)
Again, for these I'd have to find a common denominator, correct? And than multiply everything by it? So number 3 the denominator would by Y and for number 4 the denominator would be... A? Maybe I'm no where close... :?

Simplify, use absolute value notation when necessary. If it cannot be simplified, state this:
5. \(\displaystyle \sqrt{9x^2y } \sqrt{3x^5y^2}\)

6. \(\displaystyle ^3\sqrt{ \frac{8m^7}{n^2} } ^3\sqrt{ \frac{n^9}{m^2} }\)
Just to clarify, those 3's in front of the square root are supposed to be cubes, I believe, not just the number 3.
The whole square root thing I'm having a problem with. I can find the square root of simple numbers, but toss in powers and multiple numbers and I start getting completely lost.


Determine the domain for each of the following:
7. \(\displaystyle g(x) = \sqrt {5x+2 }\)

8. \(\displaystyle d(x) = -\sqrt {7x-5 }\)
These ones don't look too complicated... and yet I have no clue where to go with them. Especially the "domain" thing... What's a "domain"? I don't remember going over that in class yet...
And for number 8, I didn't think you could get the square root of a negative number? So would that simply be marked "no solution"?

I know that this is a lot of problems that I've posted, but by no means do I expect help with all of them. I only thought I'd post what I am stuck on and perhaps I could get help for what people are willing to help me with.
But I am sorry for posting so many. Believe me, I am trying my best with these, and I do have over half of the worksheet finished for my college math class, but I'm still struggling with the ones I posted. I do feel bad for posting so many... I feel like I'm asking a lot.
Like I said, I have been going to a tutor, but unfortunately it's a public tutor and there's always a large crowd. After a while I tend to get tired of waiting for forty-five minutes just to have one question quickly answered.

Err, sorry for my rambling. But any help with any of the above problems would be appreciated.

Thank you for your time.
Click to expand...

\(\displaystyle \frac{1}{x-2} + \frac{3}{x-1}\)
_______________
\(\displaystyle \frac{2}{x-1} + \frac{5}{x-2}\)

\(\displaystyle \frac{\frac{1}{x-2} + \frac{3}{x-1}}{\frac{2}{x-1} + \frac{5}{x-2}}\)

\(\displaystyle = \frac{ \frac{(x-1) + 3(x-2)}{(x-1)(x-2)} }{ \frac{5(x-1) + 2(x-2)}{(x-1)(x-2)}}\)

Now continue....
 
Blue Leader said:
Greetings,


Determine the domain for each of the following:
7. \(\displaystyle g(x) = \sqrt {5x+2 }\)

8. \(\displaystyle d(x) = -\sqrt {7x-5 }\)
These ones don't look too complicated... and yet I have no clue where to go with them. Especially the "domain" thing... What's a "domain"? I don't remember going over that in class yet...
And for number 8, I didn't think you could get the square root of a negative number? So would that simply be marked "no solution"?


Thank you for your time.
Click to expand...

I'm quite surprised that you don't "remember" seeing the definition of "domain." This is usually one of the FIRST topics covered when discussing functions.

The domain of a function is the set of all values for the independent variable for which the function is defined.

For example, suppose you have this:

f(x) = sqrt(x - 1)

In the real number system, the square root is defined ONLY for numbers greater than or equal to 0. So, if you have f(x) = sqrt(x - 1), that function is defined ONLY when the quantity under the square root sign is greater than or equal to 0.

So, the solution for

x - 1 > 0

will tell you the values for x which constitute the domain.

Try this approach on your problems.

If you're still having trouble, please follow the "rules for posting" and show us what you have tried.
 
Hi Subhotosh!
Regarding the number one:
In this excercise your best bet is factor each one of the polynomial expression, you will get a three terms one, should be able to fin a product of two Binome (I mean, something af the type (a+bx)(c+dx) and work with this)
I got:
(x[sup:4wo72omv]2[/sup:4wo72omv]-11x+28)/(x[sup:4wo72omv]2[/sup:4wo72omv]+x-30)
So you can check your work, or mine if I made a mistake
The second one:
You don't need factor anything, I got:
(4x-7)/(7x-9)
Number three:
Solve for y, try to multiply by y and you will get a second grade equation, I think is easier to factor then, but the general formula works too
I got y[sub:4wo72omv]1[/sub:4wo72omv]=-2, y[sub:4wo72omv]2[/sub:4wo72omv]=7
Number four:
Cross multiply (I mean, multiply each side by (a-7)(a+3)), then you will have a second grade equation again, again is easier to factor. I got:
a[sub:4wo72omv]1[/sub:4wo72omv]=2, a[sub:4wo72omv]2[/sub:4wo72omv]=3
Number 5:
Here both expressions have the same exponent (1/2) or Square root. Just multiply them and use exponent laws to simplify
I got: 3x[sup:4wo72omv]3[/sup:4wo72omv]y(3xy)[sup:4wo72omv]1/2[/sup:4wo72omv]
Remember that exponent 1/2 means square root, and 1/3 means cubic root
In number six I got:
2mn[sup:4wo72omv]2[/sup:4wo72omv](m[sup:4wo72omv]2[/sup:4wo72omv]n)[sup:4wo72omv]1/3[/sup:4wo72omv]
For seven and eight:
The domain of a function are the values for which the function is defined. You are right, you can only find square root of positive numbers, so, you need to find the values in each one for what the equation inside the square root is positive (Use inequalities)
For example, in number 7 the domain is every real number x that are equal or greater than -2/5. Can you find the domain for number 8?
I hope this helps.
 
I will attempt to give you some help on your Problems 7 and 8.

Problem 7: \(\displaystyle g(x) = \sqrt{5x+2}\)

The square root function is defined for all \(\displaystyle x>=0\), so we solve \(\displaystyle 5x+2>=0\). The solution is \(\displaystyle x>=-\frac{2}{5}\). Hence, the domain is \(\displaystyle [-2/5,\inf)\)
 
Hi Blue Leader,

You'll probably have better luck getting help in the future if you post each question separately.

As mmm4444bot said, there is too much material in these questions to answer them all deeply. Let me tell you what you need to learn in order to answer them:

1, 2, 3, 4:
-factoring polynomials
-adding fractions by finding the LCM
-invert and multiply rule for dividing fractions by fractions (for 1 and 2 only)

5 and 6:
-combining roots (this is easy: sqrt(2) x sqrt(3) = sqrt(2 x 3))
-multiplying and dividing exponents

7 and 8:
-definition of the domain

Try reviewing each of these topics in your textbook and come back in new topics if you have some specific questions about each.
 
Problem 8 is solved the same way.

Problem 8

\(\displaystyle d(x)=\sqrt{7x-5}\)

The domain is all x such that \(\displaystyle 7x-5>=0.\)

\(\displaystyle 7x-5=0\)

\(\displaystyle 7x=5\)

\(\displaystyle x=5/7\)

So, the domain is \(\displaystyle x>=5/7.\)
 
You must log in or register to reply here.
Top