CPhill

Note that   x^2 -7x + 10 =   (x - 2) ( x - 5)

Multiply through by the factorized form and we  have

5x + 16 =   A ( x - 5)  + B ( x - 2)

5x +  16 =   (A + B)x  +  ( -5A - 2B)

Equating terms we  see that

A + B =   5

The base of the  triangle =  5

To find the height

Area = (1/2) (5)  (height)

15  = (5/2)  (height)

15 (2/5)  = height

6  = height........this is the y interecept

Intercepts are  (-5,0) and (0 , 6)

Slope of line =  ( 6 - 0)  / ( 0 - -5)  =  6 / 5

Here's my best attempt at this one ....

There are   C(9,4)  = 126 possible sets

If we consider the sets  whose  product is NOT divisible by 4, we have the folowing  cases

Case 1  -   the set contains  four of the following six integers .... { 1, 2 , 3, 5, 7 , 9 }

Total sets possible =  C(6,4)  =  15

Case 2  - the set contains four of the following six integers  ....{1, 3 , 5, 6, 7  ,9 }

Total sets possible = C(6,4)  =  15

Total sets whose product is divisible by 4   =  126 - 2(15)  =  96

P ( that the set product is divisible by 4 )  =  96 / 126  =  16 / 21 

[ I'd like someone to  check  this answer.....!!!!  ]

Something like this (with the placement of D  inexact )

A                                D

8                                         5sqrt (2)

B               20                                   C

Draw  AC

AC  = sqrt [ 20^2 + 8^2 ]  =  sqrt [464 ]   =  4sqrt (29)

tan (ACB)  = 8/20  = 2/5

arctan (2/5)  = measure of angle ACB

Measure  of  angle  ACD = measure of angle BCD  - measure of angle ACB =    45  - arctan(2/5)

Area of figure = area of right triangle ABC  +  area of triangle ACD  = 

(1/2) (AB) (BC)  + (1/2) (AC) ( DC) sin ( ACD)  =

(1/2) (8) (20)  + (1/2) (4sqrt (29) ) ( 5sqrt (2) ) sin  [ 45  - arctan (2/5) ]   =

80  +  10 sqrt ( 58) * sin [45 - arctan (2/5) ]  =     110  units^2 

LCM of 7 , 8  =  56

The first number that they say together =  2016

The last number  that the say together = 2968

Number of integers that both say out loud =  [ 2968 - 2016 ]  / 56  +  1  =  952/56 + 1 =  17 + 1   =  18

4 (3.89) + 4(1.05) + 2(1.99) + 3 (1.79)  =   ????

2)  Note that   lxl   can be written as  sqrt ( x^2)

So

f(x) > g(x)

sqrt (x^2)  > x^2       square  both sides

x^2 >  x^4      subtract x^4 from both sides

x^2  - x^4   >   0

Note that  this is  only true  on the intervals  (-1, 0)  and (0 , 1)

Here's a graph to show this  :  https://www.desmos.com/calculator/wkih1868jq

501  503  505   ......  795   797  799

We have   [799   - 501 ]  / 2   + 1  =     (298 )/2 + 1 =  149 + 1  = 150  terms

The sum is given by

[ first term + last term ]  [number of terms ]   / 2

[ 501 + 505 ] [ 150 ]   / 2   = 

[1300 ] [75 ]  = 

97500

To average an 80   he must score a 90  on the  second exam

So....in the first case

8 =  k / 70   where k is the proportionality constant

70 * 8 =  k =  560

So....in the second  case

90 =  560 / S     where S is the hours of sleep

S  = 560 / 90  =  56/9   ≈  6.2 hrs of sleep 

Note that  OB  =OA

So triangle BOA is isosceles with angles OAB and OBA  equal

So.....measure  of angle OAB =  (180 - 113) / 2 =  67/2  =  33.5°