geno3141

Please                                   Parentheses      <---  first

Excuse                                  Exponents

My                                        Mutiplication        (this is tied with division, working from left to right)

Dear                                      Division

Aunt                                     Addition               (this is tied with subtraction, working from left to right)

Sally                                     Subtraction    <--- last

                                     20÷2(5 + 5)

Parentheses                    20÷2(10)

Division                             10(10)

Multiplication                         100

Draw right triangle ABC with the right angle at C.

If you look at angle A, since "adjacent" means "next to", there are two sides that are next to, or touching, or adjacent to angle A. One is sides AC, and the other is side AB. Since side AB alreadly has a name, it's the "hypotenuse", side AC is the adjacent side.  When looking at angle A, side BC is the side on the opposite side of the triangle, so it is called the "opposite" side.

Similarly, when referring to angle B, there are two sides that are adjacent (or touching, or next to) to angle B. One is side BC and the other is side BA. Since side BA is the hypotenuse, BC is the "adjacent" side. In this case, side AC is opposite angle B.

Once you have paid your $16.00 membership fee, you have to pay just $3.00 per visit.

So, for 10 visits, you will pay 10 x $3.00 or $30.00.

Don't forget to include your initial membership fee of $16.00,

        so your total fee will be  $16.00 + $30.00  =  $46.00.

x² - 4x + 3  =  4x - 13

Subtract  4x  from both sides:

x² - 8x + 3  =  -13

Add  13  to both sides:

x² - 8x + 16  =  0

Factor:

(x - 4)(x - 4)  =  0

So, either  x - 4  =  0  or  x - 4  =  0

--->               x  =  4    or     x  =  4

Obviously,  x  =  4.

The idea of using  a + b  and  a - b  is this:

The larger number, 206, is  a + b  and the smaller number, 194, is  a - b.

'a' turns out to be the middle number between the larger and the smaller; in this case,  a = 200  because  200  is halfways between  194  and  206.

'b' is the distance between  a  and the larger number and also the distance between  a  and the smaller number; in this case  b = 6  because  200  is  6  away from both  194  and  206.

206  =  200 + 6

194  =  200 - 6

194 x 206  =  (200 - 6)(200 + 6)

                       (a - b)(a + b)

Since  (a - b)(a + b)  =  a² - b²

             (194)(206)  =  (200 - 6)(200 + 6)  =  200² - 6²  =  40000 - 36  =  19964

Does this help?

You can solve absolute value problems with an "<" inequality in this manner:

     |x - 3| < 2

Set up this extended inequality:

   -2 < x - 3 < 2

Add 3 to all three sides:

     1 < x < 5 

(I'm going to assume that the probability of being born on any day is the same as the probability of being born on any other day and I'm going to ignore the difficulties of twin, triplets, etc. and the difficulty of having to deal with the possibility of one or more having been born on February 29.)

Let's see what the probability that all 25 will have different birthdays (which is the oppositie of what we want) and then subtract this from 1.000 to get the probability that we want.

So that all the students have different birthdays:

The first person can have a birthday on any day of the year. This person has 365 choices out of all 365 choices to have a factor of 365/365.

The second person can have a birthday on any day but the day of the first person; this person has 364 choices out of all 365 choices to have a factor of 364/365.

The third person can have a birthday on any day but the two days already taken; this person has 363 choices out of all 365 choices to have a factor of 363/365.

Notice that the numerator keeps dropping by one each time while the denominator remains at 365.

This pattern continues; the 25th person has a factor of 341/365.  Yes; it is 341/365, not 340/365; check it out!

To find the probability that all 25 persons have different birthdays, find this product:

     (365/365) x (364/365) x (363/365) x (362/365) x ... x (343/365) x (342/365) x (341/365)

Since this is the opposite of what we want, subtract that answer from 1.0000.

I wrote a compter program to evaluate this ...

I haven't found a countradiction for whole numbers up through 1000, but that doesn't prove it for all cases ...

6x + 3   >  36x + 39

Subtract 6x from both sides:

        3  >  30x + 39

Subtract 39 from both sides:

     -36  >  30 x

Divide both sides by 30:  -36 / 30  =  -6/5

     -6/5  >  x

Reversing the inequality:

          x  <  -6/5           (notice that, if you switch the x and the -6/5, you also have to switch the direction of                                                   the inequality sign)

x⁶  =  x·x·x·x·x·x

x⁴  =  x·x·x·x

So:  x⁶ / x⁴  =  (x·x·x·x·x·x) / (x·x·x·x)  =  x·x  =  x²   (Cancelling out like factors)

Or:  you can subtract exponents:  x⁶ / x⁴  =  x^{6 - 4}  =  x²