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When the value of the "solution" makes a denominator equal zero!

I have a whole section of rational equations in Intermediate Algebra. To see these, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "MATH IN LIVING COLOR." Choose "Intermediate Algebra", and look for Section 2.07 Fractional Equations. See #23 of this section to see an equation in which the solution obtained is NOT allowed.

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

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

You can put this solution on YOUR website!
Let x = weight of B
6x = weight of A

x + 6x = 154
7x=154
x=22 Meteorite A
6x= 132 Meteorite B

Check : 22+132= 154
It checks!!

For additional explanation on this type of problem, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 1 for several topics on WORD PROBLEMS. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR!

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

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You are supposed to write this in INTERVAL NOTATION.

8x-(5-x) >11x+1
8x-5+x>11x+1
9x-5 >11x+1

Subtract 11x from each side:
9x-11x-5>11x-11x +1
-2x-5>1

Add +5 to each side:
-2x-5+5 >1+5
-2x>6

Divide both sides by -2. Remember, when you divide both sides of an inequality by a NEGATIVE, you have to REVERSE the INEQUALITY:
x<-3

Interval notation is always given on a numberline from LEFT to RIGHT. This is from -inf to -3. Do NOT include the endpoints, so use parentheses instead of brackets .

Final answer: (-inf, -3)

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Dr. Robert J. Rapalje, Retired

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Do you mean |7x-4|-4=-11???

If so, add +4 to each side:
|7x-4|= -11+4
|7x-4|= -7

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

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Dr. Robert J. Rapalje

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The LCD is 6, so multiply both sides by 6.




Subtract 2x from each side:


Add +18 to each side:



Check:



It checks!!

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

You can put this solution on YOUR website!
If the question had been 12-6=n, then the answer would have been n=6.

However the question was
12-6=-n
6=-n

Divide both sides by -1:




Check:
12-6=-n
6=-(-6)
6=6 It checks!!

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

You can put this solution on YOUR website!
Apparently you purchased a set of videos from someone who advertizes on algebra.com. This website is a forum for students to post problems and free tutors to answer their questions. You should probably direct your questions to the company from which you purchased the DVDs.

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Dr. Robert J. Rapalje

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f(x)=20-2x

Just substitute 7 in place of all the x's.
f(7)=20-2*7
f(7=20-14=6

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Dr. Robert J. Rapalje

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Yes! The "letters" all cancel out, and all you are left with is the fraction . You can also write this as 1 1/3 a, but the improper fraction 4a/3 is usually preferred. By the way, you went to a lot of trouble to find all the factor combinations of 192 and 256. I think it's easier to break down a number using what is called a "factor tree." I can't really show this in the algebra.com format, but you can see what I mean on my own website.

Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 2, Section 2.02 "FACTORING NUMBERS." In this section, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks.

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

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The solution to an EQUATION is actually the endpoint of an interval that would be the solution for a corresponding INEQUALITY.

If you are asking if you can solve an inequality by solving an EQUATION and then just replace the equal sign with the inequality sign, the answer is NO! Definitely NO! The rules for solving inequalities are similar to the rules for solving equations, but definitely NOT the same. For more explanation on this, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list.

On my Homepage near the top, 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. In this section, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR!

If I misunderstood your question, please send me an Email, and I'll be glad to try again.

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

You can put this solution on YOUR website!
4x(x-2)-5x(x-1)=2
4x^2 -8x -5x^2 + 5x =2
-x^2 -3x = 2

This is a quadratic equation, so you have to set the equation equal to zero. Because you have a NEGATIVE x^2 term on the left side, it will be a good thing to get everything on the RIGHT side by adding x^2 + 3x to each side:

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

Factor the trinomial:
0=(x+2)(x+1)

Set each factor equal to zero:
x=-2 and x= -1.

For additional explanation on this type of problem, especially SOLVING QUADRTIC EQUATIONS (and maybe also FACTORING TRINOMIALS, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 2 for several topics on FACTORING and QUADRATIC EQUATIONS. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR! I also have FREE videos on the website of me teaching my classes a few years ago. To access the videos, go back to my Homepage, and look for the link "Rapalje Videos in Living Color." Select "Basic Algebra", and select the video on "FACTORING." I don't have a video on Solving Quadratic Equations.

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

You can put this solution on YOUR website!
Do you mean ? If so, remember that
means that . Likewise,
means that

6x+12=3x+4
3x= -8
x=-8/3

The check is so ugly because of the fraction, that it might be easier to just solve the problem again!

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Dr. Robert J. Rapalje

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Let x = the first number.
Let y = the different number.

4(x+5) + y
4x+ 20 +y or 4x+y+20

By the way, this is NOT an equation, and you can't solve this unless you know something more about the numbers.

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Dr. Robert J. Rapalje

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Let x = width
3x= length

2W+2L = P
2(x) + 2(3x)=320
2x+6x= 320
8x = 320
x=320/8= 40 Width
3x= 3*40= 120 Length

Check:
2*40+2*120
80+240
320 It checks!

For additional explanation on Word Problems like this, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 1 for several topics on Word Problems, especially Section 1.10 Part I that specifically covers perimeter of rectangles like this one. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR!

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

You can put this solution on YOUR website!
Is this a product of square roots? If so, then




Next, sort this out into two separate square roots to put all the perfect squares in the first square root and the leftover factors in the second square root like this:



Simplify the first square root, the second square root stays as it is!


For additional explanation on Square Roots, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 5 for several topics on SQUARE ROOTS, in particular, Section 5.01. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR! I also have FREE videos on the website of me teaching my classes a few years ago. To access the videos, go back to my Homepage, and look for the link "Rapalje Videos in Living Color." Select "Basic Algebra", and select the video on Radicals (Square Roots!).

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

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Let x= the number

6x = x+ 60

Subtract x from each side:
6x -x = x-x+60
5x=60
x=60/5=12

Check: Take the product of 12 with 6, which is 72. This is indeed the same as its sum with 60, which is also 72.

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Dr. Robert J. Rapalje

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Let x = the number
6x-4=68

Add +4 to each side:
6x=72

Divide both sides by 6:



For additional explanation on solving Word Problems, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, 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 Part I. In this and other sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR!

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

You can put this solution on YOUR website!
Let x= width of the rectangle
2x= length of the rectangle

2W + 2L = P
2(x) + 2(2x)=48
2x+4x=48
6x=48
x= 8 inches Width
2x = 16 inches Length

Check:
2*8+2*16
16+ 32
48 Checks!!

For additional explanation on Perimeter problems and other Word Problems, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, 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 Part I. In this section and others like it, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR!

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

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If surface area of a cube is 294, then remember that there are 6 faces of the cube. If you divide by 6, this means that each face is 294/6=49.

If the edge of the cube is s, then s^2=49, so s= 7.

V=s^3
V=7^3= 49*7=343

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Dr. Robert J. Rapalje

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To find the y-intercept, let x=0! Always!!! In this case, if x=0, plug it in and get 2y=10, so y=5. This is the y-intercept.

Now, there are two main ways to get the graph. Because this is in STANDARD FORM (that is, the x and y terms are on the same side of the equation!), get the x-intercept by letting y=0. If y=0, then you get x= 10. Graph these two points, connect the dots, and it should look like this:



For additional help on the topic of Graphing Lines, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 4 for the first couple of sections on Graphing and Slope of a Line. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR! I also have FREE videos on the website of me teaching my classes a few years ago. To access the videos, go back to my Homepage, and look for the link "Rapalje Videos in Living Color." Select "Basic Algebra", and select the video on "Graphing and Slope of a Line."

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

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Re-write this as
4w^2 -12w+9

Notice the perfect squares of 4w^2 and 9. This is probably a perfect square trinomial. However, even if you don't see this, just try to factor it as a TRINOMIAL:

(2w_____)(2w_____)
(2w-3)(2w-3)
(2w-3)^2

For additional explanation on Factoring, especially COMMON FACTOR and TRINOMIAL FACTORING, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 2 for several topics on FACTORING. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR! I also have FREE videos on the website of me teaching my classes a few years ago. To access the videos, go back to my Homepage, and look for the link "Rapalje Videos in Living Color." Select "Basic Algebra", and select the video on "FACTORING."

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

You can put this solution on YOUR website!
A rational expression (fraction!) is undefined if the denominator is ZERO! In this case,
q^2-4 =0
(q-2)(q+2)=0
q=2 and q=-2

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For inequalities, I recommend that you always get the variable terms on the LEFT side!

Subtract 11y from each side:



Next, get all the number terms to the right side by subtracting 24 from each side:



The last step, you must divide both sides by -18. When you divide both sides by a NEGATIVE number, you must REVERSE the inequality!




For additional explanation on Solving Inequalities, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, 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. In this section, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR!

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

You can put this solution on YOUR website!
You have to combine like terms, which means terms with the same variables raised to the same powers. You can combine the terms that contain x^3. You have 3/4 x^3 - 1x^3, which would be a total of -1/4 x^3. The rest all stays the same. Write the final answer this way:

-1/4 x^3 +4x^2 + 7

For additional explanation on the topic Combining Like Terms, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 1 for Sections 1.05 and 1.06. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR!

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

You can put this solution on YOUR website!
NO! You went TOO FAR, and you broke the LAW of MATH!!

You can multiply 1/5*1/2*3*40--these are all numbers in a product, and the result is 1/10*3*40 = 1/10*120= 12. HOWEVER, when it comes to multiplying times (s+7)^2/3, the Distributive Property does NOT apply because of the exponent 2/3. The final answer is 12(s+7)^2/3. You can't go any farther with this!!

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Dr. Robert J. Rapalje

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Rewrite this as w^2 -18w + 77
This is a trinomial, which if it factors probably looks like this:
(w_____)(w_____)

You need to find two numbers whose PRODUCT is 77, that ADDS up to -18. Try -7 times -11. It looks like this:

(w-7)(w-11)

For additional explanation on TRINOMIAL FACTORING, you have GOT to see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 2 for Section 2.04 FACTORING TRINOMIALS. I really believe I have a special explanation with repetition of exercises to help you learn this, the MOST IMPORTANT TOPIC in First Level Algebra. Honestly, this section is MUCH easier to understand than your own algebra book. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR! I also have FREE videos on the website of me teaching my classes a few years ago. To access the videos, go back to my Homepage, and look for the link "Rapalje Videos in Living Color." Select "Basic Algebra", and select the video on "FACTORING."

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

You can put this solution on YOUR website!
There are a couple of ways to do this one. I think since you have supplement AND complement in this problem, I would

let x = the angle. Then,
90-x = the complement
180-x = the supplement

Write the equation this way:

Supplement IS 4 times Complement
180-x = 4(90-x)
180-x = 360-4x

Add +4x to each side:
180-x+4x= 360-4x+4x
180+3x = 360

Subtrac 180 from each side:
3x = 180
x= 60 the angle

The complement is 90-x = 30 degrees
The supplement is 180-x = 120 degrees

Answer the question: "Find the measure of the complement." The answer to the problem is 30 degrees.

Check: "The supplement is 4 times the complement." This is true!

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

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There is a formula for the equation of a line that goes like this: y=mx+b, where m is the slope and b is the y intercept. So, if they give you the slope and y intercept, just plug the numbers into this equation:

y=mx + b
y= 1/3 x -4

For additional explanation on Graphing and Slope of a Line, please see my own website. Do a "Bing" search for my last name "Rapalje". Look for "Rapalje Homepage" near the top of the search list. On my Homepage near the top, look for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in Chapter 4 for several topics on GRAPHING. In these sections, you will find my own non-traditional explanations, that my own students all said were a LOT easier to understand than traditional textbooks. See also the corresponding "MATH IN LIVING COLOR" pages in which problems are solved IN COLOR! I also have FREE videos on the website of me teaching my classes a few years ago. To access the videos, go back to my Homepage, and look for the link "Rapalje Videos in Living Color." Select "Basic Algebra", and select the video on "GRAPHING and SLOPE of a LINE."

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

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You set it up right. Now solve for x.

180-x=2(90-x)+20
180 - x = 180 -2x + 20

Subtract 180 from each side:
-x=-2x+20

Add +2x to each side:
-x + 2x = -2x + 2x + 20
x= 20

Now, answer the question. Find one half of the complement of the angle.

The angle was 20 degrees, so the complement of this angle would be 70. Now, half of 70 is 35 degrees. Does that look right?

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Dr. Robert J. Rapalje

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Let x = smaller integer
x+2 = second odd integer (larger)

7x- 2(x+2) =201
7x -2x-4 =201
5x =201+4
5x=205
x=41 smaller
x+2 = 43 larger

Check: 7*41 - 2*43 = 201??
287-86=201 Yes! It checks!!

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