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2p+r-1=2r+p+1 can be simplified by subtracting p and r from each side.
2p+r-1-p-r=2r+p+1-p-r
p-1=r+1

Now, add +1 to each side:
p=r+2

This means that p and r are NOT consecutive integers. It does indicated that if r is even, then p is also even. If r is odd, then p is also odd. The numbers are either BOTH even or BOTH odd. However, you can't say that p is necessarily even or odd, and you can't say that r is necessarily even or odd.

The correct answer is A) NONE.

Dr. Robert J. Rapalje, Retired

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Let p=1, 2^2+1= 5 which is prime.
Let p=2, 2^4+1=17 which is prime.
Let p=3, 2^6+1=65 which is NOT prime.

The smallest positive integer p is 3.

Dr. Robert J. Rapalje, Retired

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Using 49 as a denominator, the fractions 3/49, 4/49, and 5/49 do NOT reduce. The answer is B) 49.

Dr. Robert J. Rapalje, Retired

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A rational expression will be undefined if the denominator is zero!! The problem here is to find all values for which the denominator is zero. (The numerator is actually irrelevant!) This means that


or

Dr. Robert J. Rapalje, Retired

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The restriction here is due to the fact that there is a square root in the function. The RADICAND must be greater than or equal to zero.
That is,



In interval notation, this is [-1,inf).

Please see my own website at "www2.seminolestate.edu/rrapalje". Look for the link on my Homepage "MATH IN LIVING COLOR", and look under either Intermediate (Chapter 5) OR College Algebra (Chapter 2).

I also have a video of my own teaching in class a few years ago. To see the videos, look for "Rapalje Videos in Living Color", and see either Intermediate or College Algebra for the topic "Functions, Domain, and Range".

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

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In this function , you have a SQUARE ROOT. The RADICAND (that is, whatever is INSIDE the square root) must NEVER be negative. The RADICAND must be GREATER than or EQUAL TO 0.

In this case, the domain is , so .

Please see my explanation of this topic on my own website. Go to "www2.seminolestate.edu/rrapalje". Look for the link on my Homepage "MATH IN LIVING COLOR", and look under either Intermediate (Chapter 5) OR College Algebra (Chapter 2).

I also have a video of my own teaching in class a few years ago. To see the videos, look for "Rapalje Videos in Living Color", and see either Intermediate or College Algebra for the topic "Functions, Domain, and Range".

I think you will find the explanations, both written and video, much easier to understand than traditional textbooks. My own students did!!

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

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f(x)=12/x-15

In the real numbers, DENOMINATORS must NOT equal zero. Since there is a denominator in this function, there is a restriction that x-15 must NOT equal zero. The forbidden value is x=15. All other values are permitted. The domain is all values EXCEPT x=15.

You really should see my explanation of this topic on my own website. Please go to "www2.seminolestate.edu/rrapalje". Look for the link on my Homepage "MATH IN LIVING COLOR", and look under either Intermediate (Chapter 5) OR College Algebra (Chapter 2).

I also have a video of my own teaching in class a few years ago. To see the videos, look for "Rapalje Videos in Living Color", and see either Intermediate or College Algebra for the topic "Functions, Domain, and Range".

I think you will find the explanations, both written and video, much easier to understand than traditional textbooks. My own students did!!

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

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The domain is all real numbers, since there are NO restrictions (i.e., no denominators, no square roots!) in the equation.

You really should see my explanation of this topic on my own website. Please go to "www2.seminolestate.edu/rrapalje". Look for the link on my Homepage "MATH IN LIVING COLOR", and look under either Intermediate (Chapter 5) OR College Algebra (Chapter 2).

I also have a video of my own teaching in class a few years ago. To see the videos, look for "Rapalje Videos in Living Color", and see either Intermediate or College Algebra for the topic "Functions, Domain, and Range".

I think you will find the explanations, both written and video, much easier to understand than traditional textbooks. My own students did!!

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

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* * = 1.1

I had to look up the conversion numbers on my own website! By the way, for additional explanation on Metric System conversions, please visit my website at "www2.seminolestate.edu/rrapalje". Near the bottom of my Homepage, look for "Topics from Geometry", then click on "Metric System." For all the algebra in my teaching and on this website, strangely enough, this Metric System page (that I seldom make use of myself!) is my most frequently visited page!!

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

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A vertical line is always in the form x = a number. In this case it is x=5.


Dr. Robert J. Rapalje, Retired

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I get a remainder also! Are you sure you copied the problem correctly? If the problem had been (3x^2-7x+2)/(3x-1), then the remainder is zero. Actually, the numerator would have factored, and the fraction would have reduced.

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

P.S. Check out my website at "www2.seminolestate.edu/rrapalje". It has a lot of algebra help and great joke page!!

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log 1 = 0, since 10^0=1.

Therefore, 2 log 1=2*0=0.

Dr. Robert J. Rapalje, Retired

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9

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The vertical asymptote is where the denominator equals zero! In this case, the denominator equals zero at x=6. This is the vertical asymptote.

Dr. Robert J. Rapalje, Retired

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(-2*y)^4
(-2)^4*y^4
16y^4

Dr. Robert J. Rapalje, Retired

P.S. If you need more help with the Laws of Exponents, I have a pretty good explanation on my website at "www2.seminolestate.edu/rrapalje". If you find this page, then choose the link from my Homepage "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 2 for Sections 2.12 and 2.13 on the Laws of Exponents. I think you will find it "user-friendly" and MUCH easier to read than traditional textbooks. It's all FREE.

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Let x = width
Let L = Length
P = 6x+6

2x+2L= P
2x+2L= 6x+6, you must solve for L.

Subtract 2x from each side.
2x-2x+2L = 6x+6-2x
2L= 4x+6

Divide both sides by 2.
L= 2x+3

Dr. Robert J. Rapalje

P.S. Please visit my website at "www2.seminolestate.edu/rrapalje". It's a pretty good website with lots of good math materials and a nice JOKE page!! I need the hits!!

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Area =

x= 22.36 approximately

Therefore, the length of each side is between 22 and 23.

Dr. Robert J. Rapalje, Retired

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Divided By

Factor, Factor, Factor, and INVERT the second fraction and MULTIPLY!
*

Divide out the factors (x-7),(x+2), and (x-6).



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

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They are points on the line whose equation is y = 5 .

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Is this what you mean?


If so,



Dr. Robert J. Rapalje, Retired

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The square root of 42 does NOT simplify, since no perfect squares divide evenly into it.

The only simplification that can be done is



That's it!!

Dr. Robert J. Rapalje

P.S. I have a GREAT website with a LOT of different help with square roots. Go to "www2.seminolestate.edu/rrapalje". To see my FREE video, look for the link that says "Rapalje Videos in Living Color" and choose "Basic Algebra". Select the topic "Radicals".

For written explanation, look at the link on my Homepage called "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look at Chapter 5 on Radicals. There are a LOT of problems solved in LIVING COLOR.

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The "abscissa" refers to the x-axis. The "ordinate" refers to the y-axis.
If these are "equal", then you have a line whose equation is y=x.

Dr. Robert J. Rapalje, Retired

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Divide both sides by -5, but remember that when you divide both sides by a NEGATIVE number, you REVERSE the inequality!!



Dr. Robert J. Rapalje, Retired

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Perimeter = 2W + 2L
P= 2(2x+1)+2(4x-1)
P=4x+2+8x-2
P=12x

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

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For the answer to this, you really need to see my website at www2.seminolestate.edu/rrapalje. If you can find this website, look for the link called "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "College Algebra", "Chapter 4", and look at Section 4.04 Equation Solving.

In addition to this, I have a complete explanation of Logarithms on Video that you can see for FREE from the website. On the Homepage, look for "Rapalje Videos in Living Color." Choose "College Algebra," and then look for "Logarithms."

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

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Let x = rate of the plane in still air.
Let y = rate of the wind.


Rate WITH the wind =
Rate AGAINST the wind =

x+y=600
x-y=500

Add the equations together:
2x=1100
x=550 mph = speed of the plane in still air.

Now, x+y=600, where x=550
550+y=600
y=50 mph = speed of the wind.

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

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I'm not quite sure about the wording of the last part of this, but I can tell you that for three consecutive ODD integers, you should use

x= first
x+2= second
x+4= third

The equation will be
2(x+4-6)= x+2(x+2) - 23

See how that works for you.

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

P.S. If you want to check the wording of the last part and have me finish this for you, then send an Email to Email address at rapaljer@seminolestate.edu. For some reason, the "Thank You" notes apparently are going into the spam filter tonight.

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3x+2y=18

The x intercept is where y=0. If y=0, then 3x=18, so x=6.

The y intercept is where x=0. If x=0, then 2y=18, so y=9.

The point (3,-2) means that x=3 and y=-2.
If x=3 and y=-2,
3x+2y
3(3)+2(-2) does NOT equal 18, so NO it is NOT a point on the line.

Please see my own website for additional explanation on Graphing. Do a Google search for my name "dr robert j rapalje," and look for "Rapalje Homepage" near the top of the list. Then find the link "Basic, Intermediate, and College Algebra: One Step at a Time." Select "Basic Algebra," and look in Chapter 4. You will find a complete explanation with examples and exercises, with many of the exercises solved in the MATH IN LIVING COLOR pages that are attached.

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

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The total surface area consists of the lateral (side) area and two end pieces (circles).

Lateral area is actually the distance around the base (circumference) which is times the height which is h.
Lateral area =
Lateral area =

Each end piece is a circle whose area is

The grand total =
Total Area =

Please see my own website for additional explanation on Geometric problems. Do a Google search for my name "dr robert j rapalje," and look for "Rapalje Homepage" near the top of the list. Then look for the link called "Topics from Geometry" near the bottom of the page. Several sections might be helpful to you, and it's all FREE!!!

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

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Let x= width
3x+2 = length

The formula for Perimeter is
2W + 2L = P
2(x)+2(3x+2)=14
2x+6x+4=14
8x+4=14
8x=10
x=10/8=5/4 cm
Length = 3x+2
Length = 3(5/4)+2
Length = 15/4 +2 =15/4 + 8/4=23/4 cm

Check: 2W + 2L
2(5/4) + 2(23/4)=10/4 + 46/4= 56/4= 14 cm. It checks!!

Please see my own website for additional explanation on perimeter problems. Do a Google search for my name "dr robert j rapalje," and look for "Rapalje Homepage" near the top of the list. Then find the link "Basic, Intermediate, and College Algebra: One Step at a Time." Select "Basic Algebra," and look in Chapter 1 for Section 1.11. You will find a complete explanation with examples and exercises, with many of the exercises solved in the MATH IN LIVING COLOR pages that are attached.

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

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Break it down instead of multiplying it out.


Multiply the numbers that are OUTSIDE the square root and keep them OUTSIDE the square root. Multiply the numbers that are INSIDE the square root and keep them INSIDE the square root, as follows:




For additional explanation on square roots, please see my own website. You can find my website by doing a Google search for my name "Dr. Robert J. Rapalje". Look for "Rapalje Homepage", which should be the first item on the search. Once you find my homepage, for a written explanation, click on "Basic, Intermediate, and College Algebra: One Step at a Time." Look in Basic Algebra in Chapter 5 for an entire chapter on Square Roots. There you will find a detailed explanation, and solutions to problems in LIVING COLOR!!

In addition, there is a FREE video of me teaching this topic a few years ago. To see the video, from my Homepage, look for "Rapalje Videos in Living Color." Choose Basic Algebra, and look for the video called "Radicals."
I have many other topics explained in this way from Basic, Intermediate, and College Algebra. Please check out my website to see what I have. It's all FREE.

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