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1  is in the domain of both functions....so....subbing any number in the domain into both fumctions should make them equal

So

[ 2 (1) - 8 ] / [ 1^2 - 6 - 8 ]   =  ???     2 / [ 1 - 6]

[ - 5 ] / [ - 13 ]  =  ???  2 / -5

5/13  =  ???  - 2/5

Not true...so......these functions are not equal

If the combination  can't start with  0

1 / [ 9 * 10^4 ]   =   1 / 90000

If  it can start with 0

1 / 10^5     =   1 / 100,000

(a)    P(jack and then red) 
 There are 4 jacks  and  26  reds

The probability =   4/52 * 26/52  =  1/13 * 1/2  =  1 / 26

(b)    P(red and then diamond) 

We have  26 reds  and 13 diamonds

Probability =  26/52 * 13/52  =  1/2 * 1/4  =  1 / 8

(c)    P(three and then face card) 

The are 4 threes  and 12 face cards

The probability  =   4/52 * 12/52  =   1/13 * 3/ 13  =   3 / 169 

(d)    P(ace of hearts and then eight) 

We have 1 ace of hearts and 4 eights

The probability  =    1/52 * 4/52   =  1/52 * 1/13   =  1 / 676

If B'  were (-2, -2)...... A'B'C'  would be  ABC  translated down 5  units...but since  B' = (-2,- 6) 

There is no sequence of rigid motions that maps  ABC to A'B'C'    [ last answer ]

√ [3x+1 ]   - x + 3 = 0     rearrange  as

√[3x+1]   =  x - 3       square both sides

3x + 1  =  x^2 - 6x + 9      subtract 3x , 1 from both sides and we get

x^2 - 9x + 8  =  0    factor

(x - 8) ( x - 1)  = 0      set each factor to 0  and solve for x and we get

x = 8    and   x  = 1

Check that the first answer makes the original equation true, but not the second

So....  "D"  is correct

1 / [ x - 1 ]  + 2 /x  =  x / [ x - 1]

get a common denominator  on both sides

[ x + 2 ( x - 1) ] / [ x (x - 1)]   =  x^2 / [ x (x - 1} ]

[3x - 2 ]  / [ x^2 - x ]   =  x^2 / [ x^2 - x ]

Solve for the numerators

3x - 2  = x^2     rearrange as

x^2 - 3x + 2  =  0     factor

(x - 2) ( x - 1)  =  0      set each factor to 0 and solve for x and we get the possible solutions

x  = 2    or x =1

Reject the second because it makes an original denominator  = 0  [ extraneous solution ]

"C"  is correct

The denominator cannot  = 0

So...set the denominator  = 0  to find the x values that are excluded from the domain

So we have

x^2 + 2x  - 3   =  0      factor

(x + 3) ( x - 1)  =  0

Set  each factor to 0 and solve for x  and we get that

x = -3     and  x  = 1

And these are the two values excluded from the domain

[ Second and third answers  ]

a.     4  of 10 are experienced  ....so.....4 / 10

b.    Final ratio of experienced to total players  =  [ x + 4 ]  / [ x + 10 ]

c.      Final ratio of experienced to total players =  x + 10

Here is a proof

Angle DEF  =  angle ABC              

A transversal cutting two parallel lines makes alternate exterior angles equal

Sum of  measures of angles 1 and 2  = 50° + 48°  =  98°   = measure of angle DEF  =  measure of angle ABC

But since  s bisects   angle ABC, then angle 4  =  98 / 2   = 49°

But angles 3 and 4  are vertical angles....therefore.... measure of angle 3  =  49°

Don't know much about this, RP....but....see if this diagram might give you some clues

Thanks!

We have

1/2  [  6    11      -3  ]

                  3       7 

         _____________

          6     14      4

The remainder is   4

There are  99,999 - 10,000 + 1  = 90,000 five digit numbers

This equals   90,000  less the  integers that contain no 2's or 3's

We have  7 ways to pick the leading digit  [ since 0 cannot lead ] and 8 ways to pick  each of the other 4 digits

So      7 * 8^4  =  28,762

So   90,000 - 28,762  =  61328  integers

3+6n<-5n-8<=8-3n

We have these two inequalities

3 + 6n < -5n - 8                  and      -5n - 8 ≤ 8 - 3n

                         add 5n to both sides 

3 + 11n <   -8                                  -8 ≤ 8 + 2n

subtract 3 from both sides            subtract 8 from both sides

11n < -11                                         -16 ≤ 2n

divide both sides by 11                divide both sides by 2

n < -1                                               -8 ≤ n

So....The interval that solves this is   [ -8 , -1 )       

Thank youuu!

Generally speaking, projectile motion is described by a PARABOLA.   That is, the projectile follows a parabolic path.

Straight up, the path will be a straight line.  The initial velocity will be the greatest and the object will experience a NEGATIVE acceleration (due to gravity   (sometimes air friction is considered)) until it is 'caught' or it ceases vertical motion and begins it's descent back along the line....this time being ACCELERATED by gravity.

10^(2lg a) = (10^(lg a))² = a²

I think this is supposed to be  ( ??? )

10^2 log (a)  =

100 log a  =

log a^100

x / 2  -   2 / [ x + 1 ]   =   1       get a common denominator on the left

  [ x [ x + 1 ]   - 2*2 ) / [ 2 ( x + 1) ]  =  1

[ x^2 + x - 4] / [ 2x + 2 ]  =1      multiply both sides by  2x + 2

x^2 + x  - 4 =  2x + 2      subtract  2x, 2 from both sides

x^2 - x - 6  =  0     factor

(x - 3) ( x + 2)  = 0      

Set both factors to 0 and solve for x   and we get that

x = 3   or  x =  -2

8 ( x + 3)^2            8      ( x + 3)^2

__________     =  _    * _______   =     4 (x + 3)  =   4x + 12

2  ( x + 3)              2       ( x + 3)

Expand the following:
(2 x + 1) (x - 2) (x + 3)

Multiply x - 2 and x + 3 together using FOIL.
(x - 2) (x + 3) = (x) (x) + (x) (3) + (-2) (x) + (-2) (3) = x^2 + 3 x - 2 x - 6 = x^2 + x - 6:
x^2 + x - 6 (2 x + 1)

Multiply out (2 x + 1) (x^2 + x - 6).

2x^3 + 3x^2 - 11x - 6

Thanks, Knockout...I have made the correction !!!

20^4 - 19^4 =29,679 - Four-letter words

20^3 - 19^3 =1,141 - Three-letter words

20^2 - 19^2 =39 - two-letter words.

20^1 - 19^1 =1 - one-letter word.

29,679 + 1,141 + 39 + 1 =30,860 - total number of words in Kobish alphabet.