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Divide by 7, and see what the remainder is.

327 divided by 7 gives you 46 with a remainder of 5, right?? so 327(mod 7) = 5.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Were you given the mean and standard deviation of the population???

Dr. Robert J. Rapalje, Retired

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I LOVE factoring, but I think you copied this one wrong!!! If I am wrong, please send me an Email (or a Thank You note!), and I'll try again for you.

This is a "Factoring by Grouping" problem where you must group the first two and the second two together!

x^3 - 2x^2 - 16x + 32 = 0

x^2(x-2) - 16(x-2) = 0

Now, take out the common factor of (x-2).
(x-2) (x^2 - 16) = 0

Now, factor the difference of squares:
(X-2)(x-4)(x+4) = 0

Finally, this is an EQUATION, so solve the equation by setting each factor equal to zero:
x= 2, x=4, x=-4

I have a LOT of explanations, examples, and FULL COLOR solutions to problems like this on my own website. I even have VIDEOS of me teaching this type of factoring in my younger days before I retired and my hair was still dark!! This is a GREAT chapter if you get a good explanation of it--not nearly as hard as your textbook is making it.

My website is all FREE with NO ADVERTISING, and I'm not selling anything! I just want to make this as easy as I can for you.

To see my website go to www.mathinlivingcolor.com, named after my theme "Math in Living Color." Look for a single link near the bottom of this page that will take you to my Homepage. Once on my Homepage, look for "Basic, Intermediate and College Algebra: One Step at a Time", and select "Basic Algebra". Look in "Chapter 2" for the appropriate topics!

As for the videos, look for the link on my Homepage called "Rapalje Videos in Living Color", and choose either Basic or Intermediate Algebra. Choose the videos that have to do with Factoring.

If you have questions on other topics, just send me an Email! I'll be glad to help you find the right page on my website. I hope you like my website and that you will recommend it to family and friends! Remember, my goal is to make math EASIER than your own textbook is making it, and it's ALL FREE!!!


Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

www.mathinlivingcolor.com.

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Let x = number of teams in the league. Since each game includes 2 teams, you are looking for a COMBINATION problem, C(x,2) = 45.

In other words,



Now, if you are an algebra student, you should multiply out the left side to obtain x^2 - x = 90, and solve the quadratic equation.
x^2 - x - 90 = 0
(x-10)(x+9)=0
x= 10 or x=-9, reject the negative answer. There are 10 teams in the league.

If you are a statistics student, you should ask yourself, what number times one more that the number would give you a product of 90. Probably pretty obvious that the numbers would be 9 and 10.

To analyze the answer, what you have is 10 teams in the league, and each team has 9 opponents. Lest you think this gives you 90 games, remember that Team A playing Team B is the same as Team B playing Team A, so you have to divide by 2 in order to eliminate the duplication of games.


Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

P.S. Check out my website for full color explanations and lots of practice tests with detailed solutions for all levels of math at www.mathinlivingcolor.com.

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These are NOT nearly as hard as they look!!

means to take the CUBE ROOT of (1/8), which is 1/2.
Then RAISE to the 2 power, so the answer is

means to take the CUBE ROOT of 125, which is 5.
Then, raise to the -2 power, so the answer is .

For a more complete explanation of this topic in my own words, please see my own website, which is easy to find by using the link www.mathinlivingcolor.com. Click on the link near the bottom of this page, and go to my Homepage. Once on my homepage click on the link "Basic, Intermediate, and College Algebra: One Step at a Time". Select "Intermediate Algebra", and look in Chapter 3, Section 3.05 Fractional Exponents." You probably will like my FREE explanations with many solutions in COLOR on the corresponding "Math in Living Color" pages.

I hope you like the website and that you will recommend it to friends and family.


Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

www.mathinlivingcolor.com.

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I'm sorry but I can't see how it could be 80 minutes, and I don't see anything wrong with your equation.

Let x = time it takes for the pool to fill given that the drain is open and the hose is turned on.

1/15 = amount of the pool filled in 1 minute with hose only and drain closed.
1/20 = amount of the pool emptied in 1 minute with the drain open and no hose.

, solve for x. The LCD is 60x, and the answer is x=60.

Why don't you re-post the problem. Maybe someone else has another idea on this one.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

P.S. Check out my website for full color explanations and lots of practice tests with detailed solutions for all levels of math at www.mathinlivingcolor.com.

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Converting f(x) to g(x) will be a similar process, somewhat in reverse!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

P.S. Check out my website for full color explanations and lots of practice tests with detailed solutions for all levels of math at www.mathinlivingcolor.com.

You can put this solution on YOUR website!
Take HALF of the coefficient of y, which is 7, and SQUARE it to get 49. Add this to each side of the equation:

y^2 + 14y + 49 = 32 + 49
(y+7)^2 = 81

Take the square root of each side:



Calculate each value:
y=-7 +9 =2

y=-7-9 = -16

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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DOMAIN is one of my favorite topics! It's not nearly as hard as it looks. You just have to know what is NOT allowed in math, and in this case, what is not allowed is DIVISION BY ZERO. Anytime you have a denominator, you must NEVER divide by zero, so in this case, the denominator of x-7 must NOT equal zero, so x cannot equal 7. Domain is ALL VALUES OF X except 7.

For additional explanation on DOMAIN, please see my own website. Look for the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of this page is a link that takes you to my Homepage. Near the top of the Homepage, look for the link: "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Intermediate Algebra" and look in Chapter 1, or Choose "College Algebra" and look in Chapter 2.

In addition, there are VIDEOS of me teaching these lessons before I retired. The videos can be found by looking on my Homepage for the link "Rapalje Videos in Living Color." Choose either Intermediate or College Algebra and look for the video called "Functions, Domain and Range."

You will probably like my explanations, especially the PRICE of them!! They are ALL FREE!! My own students before my retirement thought my books were easier to understand than the traditional books, and many of the solutions are given IN COLOR in the attached "Math in Living Color" pages.

Most any algebra topic you need is probably there somewhere. If you need help finding the right page on the website, please send me an Email at rapaljer@seminolestate.edu. If you like my website, please recommend it to family and friends.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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When I first read this problem, I'm afraid that I interpreted incorrectly. This is actually supposed to be a wire wrapped AROUND and UP the pipe in a spiral manner (this is called a HELIX in case you want to look into this for additional explanation),

Begin by picturing a vine or a wire wrapped 6 times around a tree trunk that is 20 feet high and whose circumference is 8 feet. If the circumference is 8 feet, then the DIAMETER of the trunk of the tree is 8/pi. Now, suppose you cut the tree off at the base, and lay the tree flat on the ground, and UNWRAP the vine or wire by fastening the vine at the top of the tree and rolling the tree on the ground. When the vine is completely unwrapped, the vine will form the hypotenuse of a right triangle, where the first leg (the height of the triangle!) is the original position of the tree on the ground, which is 20 feet, and the second leg (the base of the triangle!) is the path traced out by the base of tree as it was rolled to unwind the vine. Can you picture this?

The tricky part to figure is the length of the second leg of the right triangle. Each time you rolled the tree to unwrap one complete wrapping of the vine, the tree rolls ONE circumference, which would be 8 feet. Since the vine wrapped a total of 6 times around the tree, this means that the total distance the tree rolls to completely unwrap the vine will be 6*8 feet or 48 feet.

So, picture a right triangle whose height is 20 feet and whose base is 48 feet. The total length of the vine is the hypotenuse, which is

Of course, change the feet to cm for this particular problem. The final answer is .

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Kurt earns $60/2 units of time = $30/ unit of time compared to Paul who earns $40 per unit of time.

The ratio of Kurt's pay to Paul's pay = 30/40 = 3/4.

For additional explanation on Ratio and Proportion, please see my own website. Look for the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of this page is a link that takes you to my Homepage. Near the top of the Homepage, look for the link: "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra" and look in Chapter 1, for Section 3.08.

You will probably like my explanations, especially the PRICE of them!! They are ALL FREE!! My own students before my retirement thought my books were easier to understand than the traditional books, and many of the solutions are given IN COLOR in the attached "Math in Living Color" pages.

Most any algebra topic you need is probably there somewhere. If you need help finding the right page on the website, please send me an Email at rapaljer@seminolestate.edu. If you like my website, please recommend it to family and friends.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Let x = number of hours driving until they meet.
Distance of Sandy = R*T = 60x
Distance of Kelvin = R*T= 50x

Total Distance = 385 miles
60x + 50x = 385
110x = 385
x=385/110=3 55/110
x=3.5 hours
Since they left at 3:00 pm, add 3.5 hours, and they will meet at 6:30 pm.

For additional explanation on Word Problems, please see my own website. Look for the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of this page is a link that takes you to my Homepage. Near the top of the Homepage, look for the link: "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra" and look in Chapter 1, for Section 1.10 Word Problems.

You will probably like my explanations, especially the PRICE of them!! They are ALL FREE!! My own students before my retirement thought my books were easier to understand than the traditional books, and many of the solutions are given IN COLOR in the attached "Math in Living Color" pages.

Most any algebra topic you need is probably there somewhere. If you need help finding the right page on the website, please send me an Email at rapaljer@seminolestate.edu. If you like my website, please recommend it to family and friends.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

You can put this solution on YOUR website!
Let x = first
x+1 = second
x+2 = Third
x+3 = fourth

Second + Third = 68
x+1 + x+2 = 68
2x + 3 = 68
2x = 65
x= 32.5 Not an integer!

There is NO SOLUTION!!

For additional explanation on Word Problems, please see my own website. Look for the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of this page is a link that takes you to my Homepage. Near the top of the Homepage, look for the link: "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra" and look in Chapter 1, for Section 1.10 Word Problems.

You will probably like my explanations, especially the PRICE of them!! They are ALL FREE!! My own students before my retirement thought my books were easier to understand than the traditional books, and many of the solutions are given IN COLOR in the attached "Math in Living Color" pages.

Most any algebra topic you need is probably there somewhere. If you need help finding the right page on the website, please send me an Email at rapaljer@seminolestate.edu. If you like my website, please recommend it to family and friends.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

You can put this solution on YOUR website!
Subtract 1/4 from each side:




Draw a numberline, locate the number at m=1/4, shade everything that is to the LEFT of 1/4 and INCLUDE the endpoint.

For additional explanation on Inequalities, please see my own website. Look for the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of this page is a link that takes you to my Homepage. Near the top of the Homepage, look for the link: "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra" and look in Chapter 1, for Section 1.11 Inequalities.

You will probably like my explanations, especially the PRICE of them!! They are ALL FREE!! My own students before my retirement thought my books were easier to understand than the traditional books, and many of the solutions are given IN COLOR in the attached "Math in Living Color" pages.

Most any algebra topic you need is probably there somewhere. If you need help finding the right page on the website, please send me an Email at rapaljer@seminolestate.edu. If you like my website, please recommend it to family and friends.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

You can put this solution on YOUR website!
The key is to make sure the units in the problem are the same throughout the problem. Since you have 35 mph, you need to change the 12 minutes to 12/60 = 1/5 hour.

Now, use the famous formula D=RT, where R=35 mi/hr and T=1/5 hour.

D=RT
D=35*1/5 = 7 miles

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

You can put this solution on YOUR website!
For additional explanation on this topic, please see my own website. Look for the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of this page is a link that takes you to my Homepage. Near the top of the Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose Basic Algebra, look in Chapter 4 for Section 4.08 Applications of Systems of Equations. Several examples are solved IN COLOR, which you should find helpful.

If you like the website, please recommend it to your friends and family.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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You multiply the numbers of outcomes for each category:

Main Dishes * Appetizers * Salads * Desserts
4*7*2*4 = 224

For additional explanation on this topic, please see my own website. Look for the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of this page is a link that takes you to my Homepage. Midway down my Homepage look for the link "Topics from Probability." Everything is FREE!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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sqrt 2 has the smallest coefficient AND also the smallest radicand. Therefore it must be the smallest quantity.

For additional help with a variety of algebra topics, please see my own FREE website which can be found at the easy-to-remember and easy-to-spell link: www.mathinlivingcolor.com. Click on the link near the bottom of the page, which will take you to my Homepage. Near the top of the papge, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time."

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Let x=the angle
90-x=the compliment
180-x=the supplement

90-x + 180-x = 200
270 - 2x =200
-2x = -70
x=35 The angle
90-x =55 The Complement
180-x = 145 The Supplement

Check: 55 + 145 = 200 It checks!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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In order to get the UNION of two sets, you can ADD the number of members of the two sets together, and SUBTRACT the OVERLAP. The OVERLAP is actually the INTERSECTION. This gives the following formula:

n(A) + n(B)-n(A intersection B)=n(AUB)
20 + n(B) -16 = 58
4 + n(B) = 58
n(B) = 58-4 = 54

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Multiply both sides of the equation by the denominator 2h:


To solve for h, divide both sides by 2S:


For additional help with Literal Equations, please see my own FREE website which can be found at the easy-to-remember and easy-to-spell link: www.mathinlivingcolor.com. Click on the link near the bottom of the page, which will take you to my Homepage. Near the top of the Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose Intermediate Algebra, and look in Chapter 3 for the section on Literal Equations.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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~aVp

For additional help with Logic, please see my own FREE website which can be found at the easy-to-remember and easy-to-spell link: www.mathinlivingcolor.com. Click on the link near the bottom of the page, which will take you to my Homepage. About halfway down the Homepage, look for the link "Topics from Logic."

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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If the hypothesis of an if-then statement is FALSE, then the truth value of the statement is ALWAYS TRUE!!

For additional help with LOGIC, please see my own FREE website which can be found at the easy-to-remember and easy-to-spell link: www.mathinlivingcolor.com.

There is an entire section on "Topics from Logic" that may be helpful.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Since 45 minutes is BETWEEN 0 and 60 minutes, you use the first half of the formula, which is a flatrate $50. However, if the computer repairman runs OVER 60 minutes, then you use the rate of 4/3 x - 30. In other words, if it takes more than 60 minutes, they charge EXTRA, depending upon how long it takes. For example, if it takes 66 minutes, then you calculate 4/3*66-30, which is $58.

For additional help with most algebra topics, please see my own FREE website which can be found at the easy-to-remember and easy-to-spell link: www.mathinlivingcolor.com.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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The value of AB is 10A + B, while the value of BA is 10B + A.

In the equation, AB + BA = 1A4, substitute the values of AB and BA above:
(10A + B) + (10B + A) = 100 + 10A + 4

Subtract 10A from each side:
B+ 10B + A = 104
11B + A = 104

Subtract A from each side:
11B = 104 - A

Since B must be an integer, this means that you must subtract an integer value of A from 104 and obtain a multiple of 11.

If A = 1, you have 11B=103. 103 is not a multiple of 11.
If A = 2, you have 11B=102.
If A = 3, you have 11B=101.
If A = 4, you have 11B=100.
If A = 5, you have 11B=99, so B = 9. It works!!

Check: AB= 59, BA= 95,

AB + BA = 1A4
59 + 95 = 154, so it checks!!

Congratulations on a NICE PROBLEM!! I liked this one VERY MUCH!!

For additional help with this topic and other algebra topics, please see my own FREE website at www.mathinlivingcolor.com.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Area of three circles =



Area of unknown circle =

Divide both sides by pi:





For additional help with this topic and other algebra topics, please see my own FREE website at www.mathinlivingcolor.com.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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The point (1,-3) means to substitute x=1, and see if y=-3.

y=-2x-1
y=-2(1)-1
y=-2-1=-3. YES!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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Since you are choosing members of a team (the key word here is "CHOOSE"!), it is a COMBINATION problem, a combination of 5 people, taken 2 at a time.

C(5,2) = = 10.

Most calculators of today will calculate combinations and permuations for you.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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What was the question? Is there more information or a diagram in this problem?

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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In order to isolate the absolute value, subtract 16 from each side.
This gives


Since absolute value cannot equal a negative number, there is NO SOLUTION.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida

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For functions in which there are NO DENOMINATORS and NO SQUARE ROOTS, the DOMAIN is ALL REAL NUMBERS. An example would be y=x^2.

However, if you have a variable in a DENOMINATOR, like , then the domain is all values of x except x=5.

If you have a square root, like , then there is a restriction that the radicand, which is x-5, must be greater than or equal to zero. That is, . This means . I have a similar explanation with more details on my website.

You might want to see my own NON-TRADITIONAL explanation of FUNCTIONS, DOMAIN, and RANGE on my own website. To find my website, use the easy-to-spell and easy-to-remember link www.mathinlivingcolor.com. This link takes you to a single page with a single link near the bottom of the page. Click on this link, and it takes you to my Homepage.

Once you are on my Homepage, look for "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Intermediate or College Algebra", and look in "Chapter 5" for Intermediate Algebra, and Chapter 2 for College Algebra. Then, the hardest of the problems in these sections are solved IN COLOR on the Math in Living Color page for the respective sections.

On the topic of Functions, Domain and Range, I also have VIDEOS of me teaching this topic a few years before I retired. To find the videos, look on my Homepage for "Rapalje Videos in Living Color." Look in Intermediate or College Algebra respectively for the video on Functions, Domain, and Range. The videos, like everything else on my website, are all FREE!!! Check it out!! Math is usually not as hard as it looks if you get a decent explanation of it!!

I hope this is helpful. For ANYONE interested in using my FREE website for ANY TOPIC in math, just send me an Email at rapaljer@seminolestate.edu.

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