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Something like this     

       C  

             2    E    

    9                                   18

B            9             D              15                  A

Law of Cosines

9^2  = 24^2 + 20^2  -2 (20) (24)cos (CAB)

[9^2 - 24^2 - 20^2] / [ -2 (20) (24) ] = cos (CAB)

179/192  = cos (CAB)  = cos(EAD)

And again

ED^2  =  15^2 + 18^2  - 2 (15)(18) cos (EAD)

ED^2  =  15^2  + 18^2  - 2 ( 15)(18)(179/192)

ED^2  = 729 / 16

ED =  sqrt (729)/ 4  

ED  =  sqrt (3^6) / 4

ED  = (3^6)^(1/2) / 4

ED  = 3^3  / 4

ED =  27/4  =  6.75

                     Beret        Captain's     S of L      Mouse Ears

Slippers          a                b                c                  d

Flip-flops        e                f                 g                  h

Taps               i                 j                 k                   l

We have  12 dfferent  letters

So

We have 12 different  combinations  !!

3n + 8 < 30            2n  - 9  >   -4

3n < 22                  2n >   5

n < 22/3                  n > 5/2

The integers  that  satisfy this  system  are { 3 , 4 , 5 , 6 , 7}

Their product =  2520

Yep   47

(I gotta' get some  glasses  !!)

But you mean 47 right?

Ohhhhh. Thanks! It makes so much more sense now!

The star is just an operation symbol

a * b =    5 * 1

So   a =  5     and  b  = 1

So

5 star  1    =

9a + 2b  - ab  + 5  =

9(5) + 2(1) - (5)(1)  + 5  =

45  + 2  - 5 + 5   =

47

                  A

                            D

 B                                C

Angle ACB   = x

Angle ABC  = x + 37

Angle  A  =   180  - (2x + 37)  =  143 - 2x

And since angle  ABD  = angle ADB  then the measure of each of thee angles = (143 -2x)  / 2 

So angle  BDC =  180  - (143-2x)/2

And angle  DBC  = angle ABC  - angle ABD  =    (x + 37) - (143 -2x)  / 2

So in triangle BDC

ACB + BDC + DBC   =180

x  +  180 - (143 -2x)  / 2   + (x + 37)  - (143 -2x) / 2   =  180

Simplifying this we get

4x + 74 =  180

4x  = 106

x = 106/4  =  53 / 2   =  26.5°

So angle DBC  =     (26.5 + 37) - (143 - 2(26.5) ) / 2  =    18.5°

coefficent of x^3=85 

But then 5*1 would just be 5, and that's not correct. Maybe the star stands for something?

Yes, star means multiplication.

14 * 6  =  84 lbs

14 (9 + 1/8) oz  =  127.75 oz  =   127 + 3/4  oz  =  127 + 6/8 oz

plus

4 lbs +   (7 + 3/8) oz   =

131 lbs  +  (6/8 + 7 + 3/8) oz  =

131 lbs +  ( 7 + (6 + 3) / 8 ) oz  = 

131 lbs + (7 + 9/8) oz =

131 lbs + ( 7 + 8/8 + 1/8) oz  =

131 lbs + ( 7 + 1 + 1/8) oz  =

131 lbs + (8 + 1/8) oz =

   x lbs  +     y  oz

x + y   =  131  + 8 + 1/8  =   139 + 1/8  =  139 + .125  =  139.125  =  139.1  

48 ( 1.60)^4  ≈   315 apples

Let N  be the number of children

The total weight of the children in the first case - 94 lbs = the total weight of the children in the  second case - 58 lbs

85N   - 94  =   81N - 58

4N =  94  - 58

4N  = 36

N =  9

Yayyy!

We can form a right triangle with legs of 310  and ( 300 - 204)  = 96

The  top base  =  sqrt [ 310^2 + 96^2 ] =  324.5 cm  = missing side

The slope between  points on a line will be constant

So

(4 - 0) /( -3-9)  =  4 / (-12)  =   -1/3  =   slope

So

(0 - 1)  / (9 - a)  =    -1 / 3

   - 1 / ( 9 -a)  =   -1/3

This implies that     9 - a   =   3

So    a  =  6

x^2  will be positive for   all x's  except 0   (which will make 1 / x^2  undefined )

So   sqrt (1/x^2)  will always be >  0    

So ......the range  =  (0, inf )

We must first  solve this

sqrt ( x + 1)  = 2      square both sides

x + 1 =   4

x  =  3

So  ....     f ( sqrt (x + 1)) =   f (sqrt (3 + 1) )  = f(2)   =   1/ x  =     1 /  3

For this to have rational roots, the discriminant must be a perfect square

So

15^2  - 4 (2)b       must be a perfect square

225  -  8b

When b =       Perfect Square      

      7                   169

    81/8                144

      13                 121

   125/8               100

      18                   81

    161/8                64

      22                   49

   189/8                 36

      25                   25

    209/8                16

      27                     9

    221/8                  4

      28                     1

    225/8                  0

Sum  of the  positive integer "b" values  =   7 + 13 + 18 + 22 + 25 + 27 + 28  =  140

See my answer here :  https://web2.0calc.com/questions/help-please_58403

Angle P is greater than angles Q and R.....therefore.....

The exterior angle to P will have  the smallest degree measure.

Consider it case-by-case.

Case 1: 5 balls go in 1 box, and none go in the other. 

There is \({5 \choose 5} = 1 \) way to do this

Case 2: 4 balls go in 1 box, and 1 goes in the other. 

There are \({5 \choose 1} = 5\) ways to choose the balls to go in 1 box, leaving the other 4 for the other box. 

Case 3: 3 balls go in 1 box, and 2 go in the other. 

There are \({5 \choose 3} = 10\) ways to choose the balls to go in 1 box, leaving the other 2 for the other box. 

So, there are \(1 + 5 + 10 = \color{brown}\boxed{16}\) ways. 

Isn't it just 4?

SOlve both like equations, and the solutions will be in the overlapping regions.