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Let x=icas age
x+8= ayie age

x+x+8=32
2x+8=32
2x=24
x=12 icas
x+8=20 ayie

The sum of their ages is 32 so it checks!!

For additional help on WORD PROBLEMS, please see my own website for a LOT of FREE resources. Click on my tutorname "rapaljer" anywhere in algebra.com. Once you find my Homepage, look for "Basic, Intermediate, and College Algebra: One Step at a Time". There you will find my own user-friendly explanation for just about all the topics in algebra. Choose "Basic Algebra", then look in "Chapter 1" for "Sections 1.09 and 1.10".

Many of my exercises in this section are solved in the "MATH IN LIVING COLOR" sections that go with this. There you will find an explanation that will probably be a lot easier to understand than your own expensive textbooks!! My materials are all free.

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

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If the angles are in a ratio of 1:2:3, then let x = smallest angle.

The other angles are 2x and 3x respectively, so
x + 2x + 3x=180
6x=180
x=30
2x=60
3x=90

This is a 30-60-90 triangle, so the sides are in a ratio of 1:2:.

Let y= first side
2y= second side (hypotenuse of right triangle!)
= third side

Perimeter=


Therefore, y=10.
The smallest side is 10, the second side is 20, and the third side is . The perimeter checks!!

For additional help on GEOMETRY, please see my own website. Click on my tutorname "rapaljer" anywhere in algebra.com, and look for "Topics from Geometry" near the bottom of the page.
Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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Let x = width
x+44 = length

2W + 2 L = P
2(x)+2(x+44)=196
2x+2x +88=196
4x=196-88
4x=108
x=27 width
x+44=27+44=71 length

Check:
2*27+2*71
54+142
196
It checks!!

For additional help on WORD PROBLEMS, please see my own website for a LOT of FREE resources. Click on my tutorname "rapaljer" anywhere in algebra.com. Once you find my Homepage, look for "Basic, Intermediate, and College Algebra: One Step at a Time". There you will find my own user-friendly explanation for just about all the topics in algebra. Choose "Basic Algebra", then look in "Chapter 1" for "Sections 1.09 and 1.10".

Many of my exercises in this section are solved in the "MATH IN LIVING COLOR" sections that go with this. There you will find an explanation that will probably be a lot easier to understand than your own expensive textbooks!! My materials are all free.

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

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You can add the first three categories or numbers together, or it will be easier to add the last two numbers 3% +5%=8%, and subtract from 100%, which gives you 92%.

Dr. Rapalje

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hours
.85*60= 51 minutes

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

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What do you mean, like the abscissa, which is the x axis? And the ordinate, which is the y axis???

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Remember the law of exponents, when you raise a power to a power, you multiply the exponents!!




Is this what you meant? For more help with LOGARITHMS, I have a LOT of FREE resources on my own website. To find my website, click on my tutorname "rapaljer" anywhere in algebra.com. Once you find my Homepage, look for "Rapalje Videos in Living Color". Choose "College Algebra", then look for the videos "Logarithms Part I and Part II." The video, and ALL my videos, are FREE! You may need to download RealPlayer if you do not have it, but this is a FREE download as well.

I also have a complete written explanation of LOGARITHMS that is written for people who have trouble understanding math! To see this, look on my Homepage for the link "Basic, Intermediate, and College Algebra". Then choose "College Algebra" and look for Chapter 4.

There are also a lot of problems solved in my "MATH IN LIVING COLOR" pages. Again, look in "College Algebra."

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

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3y-1<8y-3

Subtract 8y from each side:
3y-8y-1<8y-8y-3
-5y-1<-3

Add +1 to each side
-5y-1+1<-3+1
-5y<-2

Divide both sides by -5, and don't forget to REVERSE the sign because you divided by a NEGATIVE number:



Set builder notation is {y|y>2/5}, which is read:
THE SET of all values of Y, SUCH THAT, y>2/5.

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

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approximately

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

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It depends upon where you live! Call it a "variable" expense!!

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RADICAL FORM ------->>> EXPONENTIAL FORM

really means .

A fourth root means "what number would you raise to the fourth power to get a particular number?" For example, the fourth root of 16 means what number would you raise to the 4th power to get 16? The answer is 2.

because
because
etc.

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

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Radicals are actually fractional exponents. For example,




etc.

So, the operations on radicals are actually based on the laws of expoents.

From your question, I can't tell exactly what level of math you are taking, but I may have something for you on my own website. Click on my tutorname "rapaljer" anywhere in algebra.com. Once you find my Homepage, look for "Rapalje Videos in Living Color". Choose "Intermediate Algebra", then look for the video "Radicals and Fractional Exponents." The video, and ALL my videos, are FREE! You may need to download RealPlayer if you do not have it, but this is a FREE download as well.

If you happen to be in a different level of algebra from this video, then send me an Email at rapaljer@seminolestate.edu with a couple of examples to indicate your level of math, and I'll help you find the right page to help you out.

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

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Do you mean or do you mean .

In the first case, the answer is 1/125, the second is 1/5.

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

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When solving a QUADRATIC EQUATION by factoring, the first step is to set the equation equal to zero! If you have an expression that is already in factored form on the left side of the equation, but it isn't equal to zero, then your first step is to multiply it out. Usually you will be using the FOIL method to multiply it out. Then you set the equation equal to zero. Then you re-FACTOR it to get it back into factored form. Does this answer your question??

I have a complete explanation on this topic on my own website!! You may find it easier to understand than your own textbook! Go to my website by clicking on my tutor name "rapaljer" anywhere in algebra.com. Then click on my website that will be given there.

From the Homepage of my website, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra". There you will find my COMPLETE curriculum for first level algebra. This topic of Quadratic Equations just happens to be in Chapter 2, so click on "Chapter 2" and look for "Section 2.09 Solving Quadratic Equations by Factoring." Look specifically at #55 and #56. If these are problems like your question, then go back to my Homepage, click on "MATH IN LIVING COLOR", "Basic Algebra", and look for "Section 2.09". There, you will find these specific problems solved IN COLOR!!

See if this helps. If you need to contact me, please send me an Email (on each page of my website!). You can comment in a "Thank You" note, but I have no way to respond to you.

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

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There is not enough information to solve this, unless you want a formula for length in terms of the other variables. Did you want a formula?

Dr. Robert J. Rapalje

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CORRECT!!!

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Rational expressions assume that "RATIOS" (i.e., FRACTIONS!) are involved. With polynomial expressions, you have only polynomials, like ax^3 + bx^2 +cx +d, etc.

Does that help???

Dr. Robert J. Rapalje, Retired

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An identity would be like three equations that are actually "identically" the same! How about

x+y+z=5
2x+2y+2z=10
3x+3y+3z=15

All three equations are actually the same! This would be an identity, right???

An additional thought comes to mind. Each of these equations represents a PLANE in 3-dimensions.

Did you notice that in the second equation, if you divide both sides by 2, you get the same equation as the first equation. Likewise, if you divide both sides of the third equation by 3, then you get exactly the same equation. So, the three equations that I gave as my example all represent exactly the SAME plane. Each equation is IDENTICALLY the same, so that's why we call this an IDENTITY.

Any solution of one equation will automatically be a solution of the other two equations as well.

Dr. Robert J. Rapalje, Retired

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This is NOT a LINEAR equation!! Because of the x^2 in the problem, it is a QUADRATIC EQUATION, which must be solved by completely different methods. The first step, you must set the equation equal to zero. In this case, subtract 3 from each side:
x^2+x-3=3
x^2+x-3-3=3-3
x^2+x-6=0

Next, you have to HOPE that the trinomial on the left side of the equation will FACTOR! In this case IT DOES!!
(x+3)(x-2)=0

Next, set each factor equal to zero, and solve:
x+3=0
x=-3

x-2=0
x=2

By the way, I think I have a fairly simple explanation of this topic that you may find easier to understand than your own textbook! Go to my website by clicking on my tutor name "rapaljer" anywhere in algebra.com. Then click on my website that will be given there.

From the Homepage of my website, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra". There you will find my COMPLETE curriculum for first level algebra. This topic of Quadratic Equations just happens to be in Chapter 2, so click on "Chapter 2" and look for "Section 2.09 Solving Quadratic Equations by Factoring." There you will find my own explanation of this topic, which I wrote especially for students who have trouble understanding math! Also, a LOT of the problems here are solved in the "MATH IN LIVING COLOR" pages that go with the sections in this curriculum.

I really believe math is NOT as hard as it looks, and if you get a good explanation, you CAN do it!! Get lots of help, and get lots of practice. If you like the website, please recommend my website to family and friends!!

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

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What was the question here?

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You are correct to divide both sides by 49, in order to set the equation equal to 1. Then, in the y portion of the equation, you have to invert the 2 coefficient. So in this ellipse, a^2=49 and b^2 = 49/2. The larger of these two is a=7, so the major diameter is 14.

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

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It may help to re-write this:

log6 -2log3 -3log2




or -log12

I have a LOT of resources about LOGARITHMS on my own website. Please click on my tutor name "rapaljer" anywhere in algebra.com. Click on my webaddress that is given there.

From my Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "College Algebra", and look in Chapter 4 for my entire explanation of LOGARITHMS. Here you will find my own curriculum, complete with explanations, examples, exercises, and ALL the answers, which my own students found much easier to understand than traditional textbooks. These lessons are supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved in COLOR!

In addition to this written explanation of Logarithms, I have TWO videos that were made in my own classes a few years before I retired. To access these videos, look for "Rapalje Videos in Living Color" on my Homepage. Choose "College Algebra" and look for the videos "Logarithms Part I" and "Logarithms Part II."

The videos and all the resources on the website are FREE. I spent my career trying to make math understandable to students who have trouble with math. I hope you will like my website and recommend it to friends who need help with math!!!

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

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Vertical asymptotes occur at a value of x where the denominator would equal zero and the numerator is NOT zero. In this case,

x^2-16=0
(x-4)(x+4)=0
x=4 and x=-4

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

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There are NO horizontal asymptotes. Horizontal asymptotes can ONLY occur when the degree of the numerator equals the degree of the denominator.

Dr. Robert J. Rapalje, Retired

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This appears to be a FACTORING problem!



This is a GROUPING problem in which you group the first two and the last two terms:


Now, take out the common factor of (b+3):


If you need an additional explanation on FACTORING, please see my own website. Click on my tutor name "rapaljer" anywhere in algebra.com. Click on my webaddress that is given there.

From my Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 2 for "Section 2.07 Factoring by Grouping". Here you will find my own curriculum, complete with explanations, examples, exercises, and ALL the answers, which my own students found much easier to understand than traditional textbooks. These lessons are supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved in COLOR!

I also have several videos from a few years before I retired. These videos can be accessed FREE from my Homepage by clicking on the link "Rapalje Videos in Living Color." There are videos on FACTORING in both "Basic Algebra" and "Intermediate Algebra."

All the videos and all the other pages on my website are FREE. I spent my career trying to make math understandable to students who have trouble with math. I hope you will like my website and recommend it to friends who need help with math!!!

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

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To determine the domain, it is essential that the DENOMINATOR NOT EQUAL ZERO! the numerator is actually irrelevant!

In this case, the forbidden values are values of x such that
x^2 +3x +2=0
(x+2)(x+1)=0
x=-2, x=-1

Domain is all real values of x, provided that x is NOT -1 or -2.

For additional explanation on DOMAIN, please see my own website. Click on my tutor name "rapaljer" anywhere in algebra.com. Click on my web address that is given there.

From my Homepage, look for the link "MATH IN LIVING COLOR." Choose either "Intermediate Algebra", and look in Chapter 5 or choose "College Algebra" in Chapter 2 for the topic "Functions, Domain, and Range." There you will find my own explanation with lots of examples solved IN COLOR!! My own students found this much easier to understand than traditional textbooks.

I also have videos from my own classes a few years before I retired that are available FREE from the website. From my Homepage, click on the link "Rapalje Videos in Living Color," and choose either "Intermediate Algebra" or "College Algebra" depending upon the level you are able to understand. Look for the topic "Functions, Domain, and Range."

I spent my career trying to make math understandable to students who have trouble with math. I hope you will like my website and recommend it to friends who need FREE help with math!!!

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

You can put this solution on YOUR website!
Let x = width
x+19 = length

The equation is 2W + 2L = P
2(x) + 2(x+19) = 106
2x+2x+38=106
4x+38=106
4x=106-38
4x=68
x=68/4=17 inches WIDTH
x+19=17+19=36 inches LENGTH

Check: 2*17+2*36
34+72=106
It checks!!!

If you need an additional explanation on WORD PROBLEMS, please see my own website. Click on my tutor name "rapaljer" anywhere in algebra.com. Click on my webaddress that is given there.

From my Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 1 for Sections 1.09 and 1.10. Here you will find my own curriculum, complete with explanations, examples, exercises, and ALL the answers, which my own students found much easier to understand than traditional textbooks. These lessons are supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved in COLOR! I spent my career trying to make math understandable to students who have trouble with math. I hope you will like my website and recommend it to friends who need FREE help with math!!!

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

You can put this solution on YOUR website!
The traditional way to solve this problem is to say,

Let x= first number
x+1= second number
x+2= third number

The equation is
x+x+1+x+2= 15000
3x+3=15000
3x=14997
x=4999

x+1=5000

x+2=5001

HOWEVER, the non-traditional way to solve this is to recognize that if the sum of three consecutive numbers is 15,000, the AVERAGE of the numbers is 5,000. The other two numbers are on either side of this: 4999, 5000, and 5001.

Dr. Robert J. Rapalje, Retired
Seminole State College

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A=L*W, so L=A/W





Dr. Robert J. Rapalje, Retired

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If you meant "A INTERSECT B" instead of A intercept B", then the answer is YES!

A intersect B is always a subset of B.

Dr. Rapalje

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Graphing method is nice if your equations are all in the form of "y=___" and you happen to have a graphing calculator. However, the disadvantage of the graphing method is accuracy, especially if the solutions do not come out even.

I like the substitution method if one of the equations is in the form "y=____" or "x=____", since it is easy to substitute this into the other equation.

If both equations are in standard form Ax+By=C, then usually the elimination method will be easiest!

For more on this topic, please see my own website. You can click on my tutor name "rapaljer" anywhere in algebra.com. Then click on the link to my website that is given there.

From my Homepage, click on the link "Basic, Intermediate or College Algebra: One Step at a Time." Select "Basic Algebra" and look in "Chapter 4" for the topic "Systems of Equations."

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