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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

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I can't "see" your picture, so I don't know what part of this is shaded. However, the rectangle inscribed inside the circle is 6 by 8, with center at the center of the circle. The diagonal of the rectangle will be the same as the diameter of the circle. By Theorem of Pythagoras, 6^2 + 8^2 = d^2, so d is 10. Therefore the radius of the circle is 5 feet.

As I said, I don't know what area is shaded, but the area of the circle is
sq. ft.

The area of the rectangle is A=LW= 48 sq. ft.

You probably have to subtract the areas to get the shaded area. My guess is that the area is sq. ft.

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

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Just combine like terms:
6x^7 -x+8 + 1x^7 + 2
7x^2 -x + 10

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

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When the ball hits the ground, h=0, so set the equation equal to zero.
-16t^2 + 96t +176=0
-16(t^2 - 6t - 11)=0

What results here is a quadratic equation that cannot be factored. You will have to use the quadratic formula or completing the square. It turns out that completing the square is easier for this problem.

t^2 -6t -11=0
t^2 -6t +_____= 11+_____
Take HALF of the coefficient of t, which is -6 and SQUARE it. Half of -6 would be -3, and -3 squared is +9. Add =9 to each side:
t^2 -6t+9 = 11+9
(t-3)^2 = 20

Now you can take square root of each side like this:


Then add +3 to each side:


Reject the negative value since time cannot be negative.


Finally, either calculate this, or simplify the square root. By approximation (my calculator is in the other room!) square root of 20 is about 4.5, so the time is about 3+4.5= 7.5 seconds--very approximately!!

The exact value is this:



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

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To simplify square root numbers like , you need to find a perfect square that is a factor of 44. This would be 4. By the Product Property of Square Roots, you can write this:





For additional explanation on SQUARE ROOTS, please see my own website. Use the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. Click on the link near the bottom of this page, which will take you to my Homepage. 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 sevearl topics on SQUARE ROOTS. For help with this type of problem, look in my 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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Not enough information is given to find the area of the parallelogram.

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

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Let x = distance of the train from the intersection. The distance of the car from the railroad crossing is 30 meters, and the direct distance from the car to the train is 50 meters. Notice that this picture (if you can see it!) is a right triangle, with the hypotenuse = 50 meters, the legs being x and 30.

By the Theorem of Pythagoras, a^2 + b^2 = c^2, where c is the hypotenuse.

x^2 + 30^2 = 50^2
x^2 + 900=2500
x^2 = 1600

Take sthe square root of each side:


Reject negative answer,
meters.

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

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I think you mis-typed this equation. You have TWO equal signs. Why don't you correct and re-post. Or if you are not in a hurry for this, send me an Email, and if I'll try to get it to you next time I'm available.

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The first step is ALWAYS to take out the common factor if there is one. In this case, the common factor to each term is 3 y^3.

27y^5 - 63y^4 + 30y^3
3y^3(9y^2 - 21y + 10)

What is left is what I call "Advanced Trinomial Factoring", where the leading coefficient of the trinomial is other than 1. I use a trial and error method.

It could be one of the following possibilities to give you the 9y^2 in the FIRST times FIRST term:

3y^3(3y_____)(3y______) or 3y^3(9y_____)(y_____)
Next, you need to find two numbers whose product is 10. This is probably 5*2, but it could also be 10*1. Let's try the 5*2 combination first:

3y^3(3y_____)(3y______)
3y^3(3y-5)(3y-2)

This one works, since the LAST times LAST is +10, and the MIDDLE TERM will be -6y -15y, which is -21y. It checks!

For additional explanation on Factoring, especially COMMON FACTOR and ADVANCED 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 sevearl 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

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The solution set for a set of linear equations is usually a single point, but it may also be an entire line or NO SOLUTION. The solution set for a set of linear inequalities is usually a shaded region in the coordinate plane. In extreme cases, it could possibly be NO SOLUTION or an entire line.

You can think of the examples!!

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The thing to know here is that 0! = 1 BY DEFINITION.

So, 3!/(0!*3!) = 3!/3!=1

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Dr. Robert J. Rapalje

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5a + 3b = 1 and -5a = 7 + 4a

From the second equation,
-9a=7
a=-7/9

Back into the first equation:
5a+3b=1
5(-7/9)+3b=1
-35/9 +3b=1
-35/9+35/9+3b=1+35/9
3b=9/9+35/9
3b=44/9
b=44/27



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5a + 3b = 1 and -5a = 7 + 4a

From the second equation,
-9a=7
a=-7/9

Back into the first equation:
5a+3b=1
5(-7/9)+3b=1
-35/9 +3b=1
-35/9+35/9+3b=1+35/9
3b=9/9+35/9
3b=44/9
b=44/27



Check me out to see if I have any "math" errors here. There isn't a handy way to check this without essentially re-working it. Sorry about that!!

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Dr. Robert J. Rapalje

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Let x= speed in still water
c= speed of current
Rate = Dist/Time

x+c = rate with the stream= 500 feet/5 min
x-c= rate against the stream=360 feet/6 min

x+c=100
x-c=60

Add the equations together:
2x=160
x=80 ft/min= speed in still water

x+c=100
80+c=100
c=20 ft/min= speed of current

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Dr. Robert J. Rapalje

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(ax+by)^2 +(bx-ay)^2
(a^2x^2 + 2abxy + b^2y^2) + (b^2x^2 -2abxy + a^2y^2)
a^2x^2 + b^2y^2 + b^2x^2 + a^2y^2

Did you want to factor this? If so, do factoring by grouping:
a^2x^2 + b^2x^2 + b^2y^2 + a^2y^2
x^2(a^2 +b^2) + y^2(a^2 + b^2)
(a^2 + b^2) (x^2+y^2)

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Dr. Robert J. Rapalje

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Notice that the degree of the numerator is 9 and the degree of the denominator is also 9. Send me an Email if you need additional explanation of this!

If the degrees of the numerator and denominator are the same, then there is a horizontal asymptote at y = (ratio of the leading coefficients of the numerator and denominator. In this case, it will be y=-10/1=-10.

Horizontal asymptote at y =-10.

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Dr. Robert J. Rapalje

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If this is what you meant -3|2x-5|=15, then divide both sides by -3, and you get |2x-5|= -5. There is NO SOLUTION since an absolute value cannot equal a negative number.

If you meant -3|2x|-5 = 15, then you begin by adding +5 to each side.
-3|2x| = 20. Next, divide by -3, and you get the same result: absolute value cannot equal a negative number.

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S=(a-rl)/(1-r)

Multiply both sides by 1-r:

S(1-r) = a-rl
S-Sr=a-rl

I would add +rl to each side:
S-Sr+rl =a

Then I would subtract S and add +Sr to each side:
rl=a -S +Sr

Divide both sides by r:
l=(a-S+Sr)/r

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Set up the proportion (or ratio!)

Since means ,

means that

mi.

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2x+2(x+5)>30
2x+2x+10>30
4x+10>30
4x>20
x>5

x+5>10

The width must be greater than 5 ft, which makes the length greater than 10 ft.

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Do you mean find the perimeter of a RECTANGLE? If so, this is P=2W + 2L.

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What is the question? Find the equation of a line where m=-2 and b=0?

If so, then y=mx+b, so y=-2x.

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D=rt, so dividing by t you get




Divide out the 2 factor:
km/hr


At this rate, if D=162 km,
then
hrs

If t=8 hrs, D=rt
D=54*8= 432 km

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2x^2 -7=0
2x^2 =7
x^2=7/2



The trick now is to rationalize the denominator, by multiplying numerator and denominator by square root of 2:
*


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There is NO SOLUTION, since 3x-9y can't equal both 1 and 2 at the same time. The lines are parallel.

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Area= L*W
A = 5*10 = 50 sq. ft.

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Square root of -49 is a complex solution. If complex numbers are allowed, the result is 7i.

Negative square root of 49 is -7, a real number.

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x is the common endpoint of the sides that are equal. Therefore, the two equal angles are at y and z.

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By the additive inverse property [1+(-1)] equals zero.
Then by the identity property of addition, x(y+1)+0=x(y+1).
So, actually TWO properties are involved.

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Let x = angle B
3x= angle A, the supplement of angle B

x + 3x= 180
4x=180
x=45 degrees Angle B
3x = 135 degrees Angle A

Check: 45+135 = 180

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Dr. Robert J. Rapalje

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Let x = number of minutes
75 x = amount of water transferred

1600-75x = amount of water in the larger tank after x minutes draining
550 + 75x = amount of water in the tank being filled after x minutes of filling

1600-75x = 550 + 75x

Add +75x to each side.
1600=550 + 75x+75x

Subtract 550 from each side:
1050= 150x

Divide by 150
7=x

Final Answer = 7 minutes.

Check: After 7 minutes, 7*75 = 525 gallons of water transferred

550+525 = 1075 gallons in first tank
1600-525 = 1075 gallons in second tank

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Dr. Robert J. Rapalje