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In order to rationalize the denominator, you must multiply numerator and denominator of this fraction by .





Divide out the factor of 2 in the numerator and denominator:



If you need additional help with rationalizing denominators, please see my own website by clicking on my tutor name "Rapaljer" anywhere in algebra.com. Then look for the link on my homepage called "MATH IN LIVING COLOR." Select Intermediate Algebra, and look for Chapter 3, then the section called "Rationalizing Denominators." I think you will find my explanations which are IN COLOR helpful!!

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Dr. Robert J. Rapalje, Retired
Altamonte Springs Campus
Seminole Community College
Florida

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WHAT??? Can you clarify this please???

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6=1-2n+5
6=6-2n

Subtract 6 from each side:
0=-2n

Divide both sides by -2



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Dr. Robert J. Rapalje

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Assuming that your problem is , there are TWO restrictions here. First, since there is a square root in the numerator, the quantity inside the radical, x+6 MUST BE greater than or equal to zero.

, so .

However, there is a second restriction because of the denominator. Denominators must NEVER equal zero. Anything that causes the denominator to equal zero is ILLEGAL and must NOT be allowed. So in this case, 6+x CANNOT equal zero, so x CANNOT equal -6.

The second restriction must be applied as well as the first restriction, so therefore, the TOTAL restriction is . Final Answer!

Please see my non-traditional explanation of "Functions, Domain, and Range" by visiting my website. Go to my website by clicking on my tutor name "Rapaljer" anywhere in algebra.com, and look on my Homepage for the "MATH IN LIVING COLOR" topic. You can't miss it. Then select "College Algebra" and look in Chapter 2 for the topic "Functions, Domain, and Range". There you will find my very "non-traditional" explanation (that means "easier to understand" than your textbook that you spent so much money for!!), and it's in LIVING COLOR--and ALL FREE!!!

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Dr. Robert J. Rapalje
Seminole Community College
Altamonte Springs Campus
Florida

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Let x = the number to be added.



Remember that if then

So, means that

15+5x = 40
5x=40-15
5x=25
x=5

Check: Add 5 to the numerator of 3 which gives you 8. .

It checks!!!

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The median height is the "middle" height, when the heights are all lined up from smallest to largest or largest to smallest. If there were 5 people each of different heights lined up in order of heights, then the middle person is 145 cm with two people taller and 2 people shorter than 145 cm.

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Dr. Robert J. Rapalje
Seminole Community College
Altamonte Springs Campus

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You forgot to state the problem!

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Rationalizing the denominator means to get rid of all the square roots in the denominator. You can do this by multiplying the numerator and denominator by the same quantity, and in the case of binomial denominators, you must multiply by the same binomial as the denominator EXCEPT with the OPPOSITE SIGN. You may ask, why do we do this??? Because it works!! This will probably get ugly, so hang in there! I have a pretty good lesson for this (if I do say so myself!!)on my website! To see it, click on my tutor name "Rapaljer" anywhere in algebra.com, and look for the link on my homepage that says "MATH IN LIVING COLOR." Select "Intermediate Algebra" and look for "Chapter 3" which is on Square Roots and Radicals. Choose the topic that says "Rationalizing Denominators." Now back to the problem at hand.



You will have to multiply the numerator and denominator by
It looks like this:

*



This simplifies (but not much, but at least you eliminated the square roots from the denominator!)


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Dr. Robert J. Rapalje
Seminole Community College
Altamonte Springs Campus
Florida

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What is the question? Solve for d? Solve for g? Why don't you re-state the problem or send me an Email!
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Let x = first odd integer
x+ 2 = second odd integer
x+ 4 = third odd integer

The sum of these integers (add them up!) will be 3x + 6.

The inequality will be
64<3x+6 <74

Subtract 6 from each part of the inequality:
64-6<3x+6-6<74-6
58<3x<68

Divide by 3:

19.333. . . < x < 22.666. . .

Since these are INTEGERS you are looking for, the only ones that fit this description would be x= 20, 21, and 22. You must reject the x=20 and x=22 since these are NOT odd integers. The only solution would be

x=21
x+2 = 23
x+4=25

The sum is 69.

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Dr. Robert J. Rapalje
Seminole Community College

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log4(100) - log4(32) - log4(50)

Using the Laws of Logarithms:





Now, by the basic definition of logarithms:
means

So, means that

Therefore, y = -2, since

I think my explanation of LOGARITHMS is easier to understand than the ones given in traditional textbooks. It's all posted free for you on my website. Just go to my website by clicking on my tutor name "Rapaljer" anywhere in algebra.com, and look for the link on my homepage that says "Basic, Intermediate, and College Algebra: One Step at a Time." Select "College Algebra" and look for "Chapter 4" which is my Logarithms chapter. You will probably need to study the Laws of Logarithms, once you understand the basic Definition of Logarithms. See also my MATH IN LIVING COLOR PAGES that correspond to this sections!

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Dr. Robert J. Rapalje
Seminole Community College
Altamonte Springs Campus
Florida
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You will have to write three equations with three unknowns.

Let b = weight of baseball
f = weight of football
s = weight of soccer ball

b+f=1.25
b+s=1.35
f+s=1.9

Start by subtracting the first equation minus the second equation.
b+f=1.25
b+s=1.35

b-b+f -s = 1.25-1.35
f-s=-.1

Now take this equation and add it to the third equation:

f-s=-.1
f+s=1.9

2f=1.8
f=.9

Now go back to the first three equations:
b+f=1.25
b+s=1.35
f+s=1.9

In the first equation, substitute f =.9
b+f=1.25
b+.9=1.25
b=1.25-.9
b=.35

In the second equation, substitute b.35
b+s=1.35
.35+s=1.35
s=1

Check with the third equation
f+s=1.9
.9+1=1.9
It checks!!!

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Dr. Robert J. Rapalje, Retired
Seminole Community College
Altamonte Springs Campus
Florida

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Let x = width of the rectangle
x+5 = length

2W + 2L = P

2(x) + 2(x+5) = 58
2x+2x+10 = 58
4x+10=58
4x = 48
x=12 width
x+5 = 12+5=17 length

Check:
2*12+2*17
24+34= 58 = Perimeter

For additional explanation on solving Word Problems please see my website by clicking on my tutor name "Rapaljer" anywhere in algebra.com. On my homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Go to "Basic Algebra", and look for "Chapter 1" and select the sections marked "Word Problems." See also my MATH IN LIVING COLOR pages, where a LOT of these problems are solved in COLOR--it really makes a difference!! My students have ALL told me that my explanations are easier than traditional books. I WROTE ALL OF THESE SECTIONS SPECIFICALLY FOR PEOPLE WHO DON'T GET MATH!! I hope it helps you!!

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Dr. Robert J. Rapalje, Retired
Altamonte Springs Campus
Seminole Community College
Florida

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You forgot to state the problem!

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This sounds like fun! Get as many different combinations as you can think of, and see which ones are missing. Here are a that I could think of:

77/7-7=4
(7-7)/7 +7 = 7
((7*7+7)/7 =8
(7+7)/7 +7 =9
(77-7)/7 =10
(77+7)/7 =12
(7*7)/7+7=14
77/7+7=18
7+7+7+7 =28
7*7 +7/7 = 50
77-7/7 = 76
77+7/7=78
77+7+7=91
This sounds like a good group project or a contest to see who can come up with the most different numbers. Am I right??

Have fun with the rest of these!!

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Dr. Robert J. Rapalje, Retired
Seminole Community College
Altamonte Springs Campus
Florida

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Domain and range are both ALL REAL NUMBERS. I'm not sure by I suppose you could write this in absolute value form:

D: |x|< inf
R: |y| < inf

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We can't see the triangles!

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Let y be the angle, and
90-y = the complement of the angle.

The statement you gave is
"Y is six more than three times the complement". Just translate it into math symbols:
y= 3(90-y) + 6

Now, solve for y:
y = 270 -3y +6

Add +3y to each side and combine the 270 + 6
y + 3y = 276 - 3y + 3y
4y = 276

Divide by 4:
y = 276/4= 69 = The angle

90-y = 90 - 69=21 = The complement

Check the equation:
y= 3(90-y)+6
69= 3(21) + 6
69= 63+6
It checks!!

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Dr. Robert J. Rapalje
Seminole Community College

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To convert from millimeters to meters (smaller to larger units!) you must move the decimal 3 places to the left, which would be 0.014 meters.

The most frequently visited page on my entire website it my "METRIC SYSTEM" page. You can find it by clicking on my tutor name "Rapaljer" anywhere in algebra.com. Look for the link on my homepage that says "Geometry Topics." Select "Metric System". I think you will like my explanation on this topic.

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Dr. Robert J. Rapalje
Altamonte Springs Campus
Seminole Community College
Florida

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24a-22=4(1-6a)

24a - 22= 4- 24a

Let's get all the variables on the left side by adding +24a to each side:
24a + 24a -22 = 4

48a -22 = 4

Next, get all the number terms on the right side by adding +22 to each side:
48a -22+22 = 4 + 22
48a = 26

Finally, divide both sides by 48:


Reduce the fraction on the right side by dividing out a factor of 2:



For additional help, please see my own website by clicking on my tutor name "Rapaljer" anywhere in algebra.com. Look for the link on my homepage that says "Basic, Intermediate and College Algebra: One Step at a Time." Select "Basic Algebra" and look in "Chapter 1" for the "Equation Solving" topic. I think you will like and understand my explanation and format for this topic. I also have a LOT of problems like this solved in my "MATH IN LIVING COLOR" pages.

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Dr. Robert J. Rapalje
Altamonte Springs Campus
Seminole Community College
Florida

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Maybe the equation of best fit??? I'm really not sure what you mean.

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Let x = number of quarters
2x+10 = number of dimes

The equation is based upon the values of each coin times the number of each coin. You can do it in either dollars or cents. Your teacher may have a preference, but to me, working with dollars requires the use of decimals, whereas the use of cents comes out in whole numbers. I'll do it in "cents". Doesn't that make "cents" to you?? Of course the quarters are each worth 25 and the dimes are each worth 10 cents. Be sure to change the $37 to cents also.

25(x) + 10(2x+10) = 3700
25x + 20x + 100 = 3700
45x +100 = 3700
45x=3600
x=3600/45=80 Quarters
2x+10 = 2(80)+10= 170 Dimes

Check by calculating the values of the coins:
Quarters = 80*.25 = $20
Dimes = 170*.1= $17
Total = $37 It checks!!

For additional help with coin problems or any kind of word problems, please see my website by clicking on my tutor name "Rapaljer" anywhere in algebra.com. Look for "Basic, Intermediate, and College Algebra: One Step at a Time." Select "Basic Algebra", and look for the appropriate topic in "Chapter 1". This is a user friendly explanation with examples and exercises integrated into it. My students ALL tell me it's easier to understand than the traditional textbooks! See also my "MATH IN LIVING COLOR" pages where a lot of these exercises are solved in "Living Color."

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If this is a problem with a complex fraction (fractions within fractions!), then one of the lines must be longer than the other one. From what you have stated without any parentheses, it's not possible to know for sure what you mean. Is it

1)
=
=

or is it

2)
=
=

Do you understand the difference?

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Hi Nick!!

Given , you must solve for x by "undoing" all the operations that were done to it. You must undo mult by 2, undo the "ln" function, and undo multiplication by 4. That brings you down to x.

Start by dividing both sides by 2:


In order to "undo" the "ln", you must raise both sides as a power of e:



Last, divide both sides by 4:


In the above statements, y represents f(x).

NOW, if you interchange the x and the y, the NEW y represents the inverse function .


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Dr. Robert J. Rapalje, Retired
Seminole Community College
Florida

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3/4 divided by -1/2

Invert the second number and multiply:


You may want to just multiply numerator times numerator and denominator times denominator, and reduce the fraction:

or -1 1/2

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12-6x+3(2x+3)=2x+5
12-6x+6x+9=2x+5
21=2x+5
21-5=2x-5
16=2x
8=x

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You aren't doing anything wrong! Math is like life--sometimes it comes out even, sometimes it doesn't!! In this case the answer is a fraction!!

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Let x= the complement of the angle
8x - 9 = the angle itself

x + 8x -9 = 90
9x = 99
x=11 the complement of the angle
8x-9= 8*11 -9=88-9 =79 degrees = the angle

Check: 79+11=90

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H=p*a^t
H=10*2^3
H=10*8=80 homes

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H= p*a^t
H = 10*2^3
H = 10*8
H = 80 homes

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