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Let x = total number of apples

The first troll demands half of his apples plus 2, so after leaving the first troll, the prince has
apples


The 2nd troll demands half of what is left and 2 more, so now the prince has




The 3rd troll demands half of what is left and 2 more, so now the prince has


And this equals exactly 2 apples!


Now, work this backwards by adding +2 to each side:


Multiply both sides by 2:
(1/4)x-3=8

Add +3 to each side:
(1/4)x=11

Multiply both sides by 4:
x=44 apples.

Check: If the prince starts with 44 apples, the first troll gets 22 +2 =24 apples. The prince is left with 20. The second troll demands 10+2, which leaves the prince with 8. The third troll demands 4 apples plus 2, leaving the prince with 2 apples.

Very nice problem!! Poor princess!!

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

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r=1, h=5

=5.23 cubic inches approximately

To convert to cubic centimeters, multiply by the conversion factor 2.54^3

cubic centimeters approximately

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No problem stated???

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f(x)=-3x^2

If x=0, then f(x)=0
If x=1, then f(x) = -3(1)^2=-3
If x=2, then f(x) =-3(2)^2=-12
If x=-1, then f(x)=-3(-1)^2=-3
If x=-2, then f(x)=-3(-2)^2=-12



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Do you mean graph y=-x+3??

The y intercept is 3 and the slope is -1. Start by going UP the y-axis 3 units, and put a point at (0,3). From this point, the slope is -1/1, so go DOWN 1, go RIGHT one unit, and put the second point. Connect the dots. It should look like this:



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Odds in favor of winning =7/5

This means number of favorable outcomes is 7
Number of unfavorable outcomes is 5

Therefore the total outcomes is 12

Probability = =

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The upper limit is 4.75+.05= 4.80
The lower limit is 4.75-.05=4.70

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16x^2 -14x+1=-6x

This is a quadratic equation, so start by setting it equal to zero by adding +6x to each side:

16x^2 -14x+6x +1=-6x+6x
16x^2 -8x+1=0

This is perfect square trinomial, so it factors like this:
(4x_____)(4x_____)=0
(4x-1)(4x-1)=0
4x-1=0
4x=1
x=1/4

In this exercise, you may need additional help with solving quadratic equations. However, the underlying problem may be being able to factor this trinomial. Either way, I have pages on my website that may be helpful to you.

If you need additional help with "Factoring" or "Solving Quadratic Equations", check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Basic, Intermediate and College Algebra: One Step at a Time." Click on "Basic Algebra", and look for the appropriate sections in Chapter 2. My non-traditional explanation should be easier to understand than your textbook!

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

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Let x = age of son now
y= age of father now

x-2= age of son 2 years ago
y-2 = age of father 2 years ago

x+3 = age of father 3 years from now
y+3= age of father 3 years from now

y-2=4(x-2)
y+3=3(x+3)

y-2=4x-8, so y = 4x-6
y+3=3x+9, so y=3x+6

Now, since
y=4x-6 and y=3x+6,
4x-6=3x+6
x=12 Age of son now
y=3x+6
y=3(12)+6
y=36+6=42 Age of father now

Check:
2 years ago, son was 10 and father was 40--(four times as old!)
3 years from now, the son will be 15 and father will be 45--(three times as old!!)

I would love to recommend my website to you on this, but I just don't have anything like this on it!!!

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

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Just multiply the number of cubic feet times the weight of each cubic foot. That would be 22.5*62.4 =1406.25 pounds.

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To find the domain of a function, make sure DENOMINATORS are NOT ZERO, and RADICANDS (whatever is inside the square root!) must be GREATER THAN OR EQUAL TO ZERO!!

In this case ,
so


If you need additional help with Domain and Range, check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Math in Living Color." Click on "College Algebra", and look for "Section 2.07, Functions, Domain, and Range." My non-traditional explanation should be easier to understand than your textbook!

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

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The center is at (0,0) and the radius is r. That's it!!

If you need additional help with Circles, you may want to check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Math in Living Color." Click on "College Algebra", and look for Section 2.04 "The Circle" at this link. My non-traditional explanation should be easier to understand than your textbook!

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

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Let x = smaller piece
5x= larger piece

x+5x=72
6x=72
x=12 cm smaller piece
5x=5(12)=60 cm larger piece

Check: 12 +60=72
It checks!

If you need additional help with Word Problems, check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Basic, Intermediate and College Algebra: One Step at a Time." Click on "Basic Algebra", and you will find a couple of sections near the end of Chapter 1 on "Word Problems". My non-traditional explanation should be easier to understand than your textbook!
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Dr. Robert J. Rapalje, Retired
Seminole Community College
Altamonte Springs Campus
Florida

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If you need additional help with square roots, check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Basic, Intermediate and College Algebra: One Step at a Time." Click on "Basic Algebra", and you will find a couple of sections near the end of Chapter 5 on "Radicals". My non-traditional explanation should be easier to understand than your textbook!

See also my "MATH IN LIVING COLOR", "Intermediate Algebra", "Chapter 3".

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

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Two vertical asymptotes:
. The asymptotes are at x= 2 and x=-2

Rational function with a hole in the graph
. The hole is at x=2, asymptote at x=-2

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x-2y=1

x=2y+1

If y = -2, then x=2*-2+1=-4+1=-3
If y=0, then x=1
If y =5, then x=2*5+1 = 10+1=11
If y=8, then x=2*8+1=16+1=17

Domain = {-3,1,11,17}

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Did you leave something out of this? There must have been something about y that you didn't mention in the problem. It seems that the y came out of nowhere!

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Domain is the set of all x values, which would be {-2,0,4,7}, so y = {11,7,-1,-7}.

If x= -2, then y=-2*-2+7=4+7=11.
If x=0, then y= 7.
If x=4, then y=-2*4+7=-8+7=-1.
If x=7, then y=-2*7 +7 ==14+7=-7

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You need more information to solve this.

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For consecutive integers, use x, x+1, x+2, etc.

For consecutive even or consecutive odd integers, use x, x+2, x+4, etc.

If you need additional help with Word Problems, check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Basic, Intermediate and College Algebra: One Step at a Time." Click on "Basic Algebra", and you will find a couple of sections near the end of Chapter 1 on "Word Problems". My non-traditional explanation should be easier to understand than your textbook!

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

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e = K(D-d)/f
Since most folks don't particularly like FRACTIONS, I recommend you multiply both sides by the denominator f.

ef=K(D-d)

Next, remove the parentheses using the distributive property:
ef =KD - Kd

To solve for D, you need to get the D term alone on the right side. Do this by adding +Kd to each side:
ef+Kd = KD

Finally, to get D by itself, divide both sides by K:



By the way, don't DARE divide out the K's, since the K in the numerator is in a TERM!!

If you need additional help with Literal Equations, check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "MATH IN LIVING COLOR." The topic of Literal Equations can be found at ALL three levels. For "College Algebra" look in Chapter 1. For "Intermediate Algebra" look in Chapter 2, and in "Basic Algebra" look in Chapter 3. My non-traditional explanation should be easier to understand than your textbook!

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

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Looks right to me!!

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Let x= number of words Todd spells correctly
48-x = number of words that he misses

In order to avoid the decimals, let's solve this in "cents". Does that make "cents" to you??? It means that he gets 10 cents for correct words, he pays -6 cents for words missed, and there is a total income of 192 cents. Here is the equation:

10*x -6(48-x) = 192
10x - 288 + 6x = 192
16x- 288=192
16x= 192 +288
16x = 480
x=30 words correct
48-30=18 words missed

Check: 30*$.10 -18*$.06 = &1.92
$3.00-$1.08=$1.92
It checks!!!


If you need additional help with Word Problems, check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Basic, Intermediate and College Algebra: One Step at a Time." Click on "Basic Algebra", and you will find a couple of sections near the end of Chapter 1 on "Word Problems". My non-traditional explanation should be easier to understand than your textbook!

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

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2lnx + lny - ln(z-3)



If you need additional help with logarithms, check out my website by doing a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" at the top of the search list, and choose the link on my homepage "Basic, Intermediate and College Algebra: One Step at a Time." Click on "College Algebra", and you will find an entire Chapter, Chapter 4, on Logarithms. My non-traditional explanation should be easier to understand than your textbook!

In addition, I have two videos that are freely available on my website. From my Homepage, click on the link "Rapalje Videos", look for "College Algebra" and then "Logarithms." The videos are a few years old, and the calculator technology is out of date, but this is an explanation that should be easier for you to understand.

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

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Get the x terms on one side by subtracting 5xy from each side:

2xy=3y+5xy
2xy -5xy=3y
-3xy=3y

Divide both sides by -3y to get the x isolated:



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When there are two variables without any other conditions given, you can't solve for x and y. The best you can do is to solve for y in terms of x or x in terms of y.
4x-2y +70=180
4x-2y = 110
4x-110=2y
2x-55=y
y=2x-55

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Divide both sides by

or

Reject the negative value.

Radius = 4 cm.

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When you have a log base b of b raised to the power, the answer is always the power! That's what a logarithm is!!

Final answer is the exponent,

Please see my own website for an entire explanation and videos on logarithms.

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Approximately 2-4-6=-8

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