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If m = 5 and n = -2, evaluate 3m - 4n and n^2 - m and find the product of the two expressions.

[ 3(5) - 4(-2)]  *  [ (-2)^2  - 5 ]  =

[15 + 8 ] *  [ 4 - 5 ]  =

[23] * [ -1]  =

-23

We have the points on the graph  (1,0)  and (-1.2)

The slope  of  this line  =   [2 - 0] / [ -1 -1]  =  2/-2  =-1

So   the equation of the line is

y  = -(x -1)

y  = -x + 1

x + y   = 1

So  we either have

x + y    ≤  1     (1)         or       x  +  y   ≥  1   (2)

Note that  (0,0)  is in the shaded area....this satisfies  (1)

So....the inequality is

x + y    ≤  1

2. Lengths of newborn girls are normally distributed with a mean of 49.2 cm and a standard deviation of 1.8 cm.

Find the z-score associated with newborn girls who are 50 cm long:

&

What percent of newborn girls will be 50 cm or longer?

[ 50 - 49.2]

__________  = z score  =  .44

    1.8

% of newborns 50 cm longer    =   [1 - .6700 ]  =  .33  ≈  33%  

1. The scores on a standardized test are normally distributed with a mean of 500 and a standard deviation of 100. 

Sofia scored 632 on the test.

What is the z score associated with Sofia's score?

&

What percent of students scored below Sofia?

[ 632 - 500 ]

___________  =  z score  =  1.32

   100

% of students scoring  below Sophia  =  .9066  =  90.66%

See the following image :

Since this is a right triangle....the center of the circle will lie  on the midpoint of  hypotenuse BC

The midpoint of the hypotenuse  is   (2, 1.5)

The radius   of the circumscribing circle  is  sqrt [  ( 4-2)^2 + (3 - 1.5)^2 ]  =  

sqrt [ 2^2  + 1.5^2 ]   =  sqrt [ 4 + 2.25]  =  sqrt [ 6.25]  = 2.5

So  the equation of the circumscribing circle is  (x - 2)^2 + (y - 1.5)^2  = 2.5^2

And we can find central angle ADB  using the Law of Cosines

AB^2  = AD^2 + BD^2 - 2(AD)(BD) cosADB

4^2  = 6.25 + 6.25 - 2 (2.5)(2.5)cosADB

[16  - 2(6.25)]  / [ -2(2.5)(2.5)) ]  =  cos ADB

And  using the arccos, we can find the measure of ADB

arccos [ (16 - 2(6.25)^2) / ( -2(2.5)(2.5) ]  = ADB ≈ 106.26°

So  the  area of the  sector DAEB  = pi * (2.5)^2 * (106.26/ 360)      (1)

And the area of triangle ADB  =  (1/2)(2.5)^2 sin (106,26°)       (2)  

So  the area  between the side of 4 and the circumference  =  (1)  - (2)  =

(2.5)^2 [ pi * (106.26/360)  - sin (106.26°) / 2 ]  ≈  2.796 units^2

\(\dfrac 2 3 t - 1 < t + 7 \leq -2t+15\\ \dfrac 2 3 t -8< t \leq -2t+8\\~\\ \text{left side}\\ \dfrac 2 3 t -8 < t\\ -\dfrac 1 3 t < 8\\ t > -24 \)

\(\text{Right side}\\ t \leq -2t+8\\ 3t\leq 8\\ t\leq \dfrac 8 3\)

\(\text{final result}\\ -24 < t \leq \dfrac 8 3 \\ t \in \left(-24, \dfrac 8 3 \right]\)

\(f(g(x)) = 4g(x) + c = \\ 4(cx+2)+c = \\ 4cx+(8+c) = 12x+d\\~\\ 4c=12\\ 8+c=d\\~\\ c=3\\ d=11\)

1)  C

2)  E

3)  B

For a moment I felt that I'm in a 3rd grade again.

l 2x + 4  l  = 14       we have two equations to solve

2x + 4  = 14              2x + 4  = -14

2x  =10                      2x  = -18

x  = 5                           x = -9

So  the product of the solutions  =  5 * -9   =   -45

If f(x) = 2x^2 and g(x)= x^2 + 2x, which expression represents g(f(x))?

g (f(x))  means that we are putting  f  into g...so we have

(2x^2)^2 + 2(2x^2)  =

4x^4  + 4x^2

For  the last two, sub 3  into both functions and then evaluate each one

Really , how has it been so far. This forum can prove to be really helpful since you get kind of real life people helping you out with real time solution and also clearing your doubts….. and btw thanks..

x^2 + 10x + 2  =  0

x^2  + 10x    =   - 2

Complete the square on x

x^2 + 10x + 25   =  -2 + 25

(x + 5)^2  =  23    take both roots

x + 5  =  ±√23

x  =  -5  ±√23

 √[x + 3]     we cannot have a negative under the radical....so...undefined for  (-inf, -3 ) ⇒  A

Perimeter =   2L + 2W

Line L has equation y = 4x - 7, and line M is parallel to line L and passes through the point (2, 15). Determine the equation of line M.

A) y = 4x + 7

B) y = 2x + 7

C) y = 4x + 15

D) y = 2x + 15

E) y = - 1/4x + 7

Rectangle EFGH has the coordinates

E (-3, -1) 

F (-3, 3) 

G (-1, 3)

Find the coordinates of point H.

A) (-1, 0)

B) (-1, 1)

C) (1, -1)

D) (-1, -1)

E) (0, -1)

Which statement MUST be true given that a is positive and b, c, and d are all positive integers, and a < 1 < b < c < d.

A) 1/a > b

B) b - d > c - d

C) c + 1/a < d + 1/a

D) a > 1/d

E) c < a

First one

N ( basketball , football, both)  = N(basketball)  + N(football)  - N (both)  = 9 + 10  - 4   =   15 

Second one

10,000 to 24,999  =

%(10000 to 14999)  + % ( 15000 to 24999)  =

         18%   +  22%  =

               40%

[ 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + n]

___________________________________   =  10         

              10 

Multiply both sides by 10  an simplify

9 + n   = 100     subtract 9 from both sides

n  = 91

Well...one way to do this is just to graph this, here : https://www.desmos.com/calculator/qoifb2gezc

Note that all "odd" multiplicities of a zero will "cut" the x axis [or cross through while "hugging" ]   while all the "even" multiplicities  of a zero will "kiss" the x axis

So....for the first one  ...."d"  is true

For the second ...."b"  is true

For the last..we have one more thing that can happen with an "odd" multiplicity.."a" is true  because the zero at 2  is of an odd multiplicity > 1 .....[this will always happen when we have an "odd" multiplicity > 1 of a zero   ]

Hope that helps  !!!!

Second one

A (1, 1) 

B (2, 3) 

C (5, 3)

The first thing to do is to plot these here : https://www.desmos.com/calculator/viqqgx48qv

We can find the coordinates of D  by noting  that  we had to subtract 1 from the x coordinate of (2,3)   and 2 from the y coordinate of (2,3)  to get (1,1)

This implies that if we  subtract 1 from the x coordinate of (5, 3)  and 2 from its y coordinate we get  

(5 - 1, 3 - 2)  =  (4, 1)  =   D

See here : https://www.desmos.com/calculator/ciuabtpszj

I was unsure whith this one, but it makes sense now. Thank you!

Awesome! Thanks

The answer in both cases i 'B' 

First one

E. (2, -1)

Last one 

C)  C

First one.....this is a 3-4-5 right triangle

x = 3     x + 1  = 4      x + 2  = 5

tan A  = BC/AB  = x / [ x + 1 ]  =  3 / 4

Second one....this is a little tough

We have a semi-circle  with a radius of 3

So....its perimeter  =  pi * 3   units

Then, we have a four sided figure

Two of sides have a length of 3  =   6   units

Let the bottom vertex of the figure lie at (0,0)

Then the bottom left vertex lies at  (-5, 2)

And the length of the bottom edge  =  sqrt (5^2 + 2^2)  = sqrt(29) units

And the bottom right vertex lies at  (1,2)

So...the length of the bottom right edge  = sqrt( 1^2 + 2^2)  = sqrt(5) units

So....the total perimeter  is    [ 3pi + 6 + sqrt(29) + sqrt(5) ]  = 23.046 units

Last one

Easy 

area  =  base * height  = 14 * 10   = 140 in^2

For the triangle problem, the triangle has to be a 3, 4, 5 triangle.

therefore, tangent A = opposite over adjacent = 3/4 = 0.75 

.

The angle (ß) at point  'C'  is:  

tan ß = 4 / 6

ß = 33.69º

MN = tanß * NC

MN = 8 

That was too easy 

Second one

Again  the highest power term   -2x^5   will determine the end behavior

When  x  is a large negative   -2(-x)^5  =  will be positive

So  as x approaches -inf, f(x)  approaches infinity

On the other hand, when x is a large positive   -2(x)^5  =  will be negative

So  as x approaches inf, f(x) approaches   negative infinity

See the graph here : https://www.desmos.com/calculator/6swdftdxku