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Thanks, icecreamlover !!!!

This problem is WAY more involved than you may think!!

1/j  + 1/k   =  1/30

Let  30  =  z

j and  k  must  be >  30

So

Let   j  =  z + m      and   k =   z  +  n

So  we  have

1/ ( z + m)  +  1/ ( z + n)  =  1/z

(z +n  + z + m)            1

____________  =     ____                 cross-multiply

(z + m) (z + n)             z

(m + n + 2z) z  =    (z + m) (z + n)

2z^2  + zm + zn  =  z^2  + zm + zn  +  mn

z^2    =   mn

30^2 =  mn

900  = mn

Now....I WILL NOT list all the  integer possibilities for this problem but here's a start

mn = 900

So  we have that

m    n          j = z + m       k = z + n

1    900             31              930

2   450              32              480

3   300              33              330

4   225              34              255

.

.

.

And so on !!!!!

WolframAlpha has done the heavy lifting  here :

https://www.wolframalpha.com/input?i=1%2Fj+%2B+1%2Fk+%3D+1%2F30

P.S. - I wouldn't even attempt to sum all the positive k"s

well well well, looks like someone's back!...

Welcome back CPhill!...

Simplify as

[ 3x + 1 + 2x - 4]  / [ x + 8]   =

[ 5x - 3 ] / [ x + 8]

This function has a horizontal  asymptote at   y =   (the ratio of the coefficients on x ) = 5/1 = 5 

The range is  (-inf, 5) U (5, inf)

We can also find the area of the larger rectangle we can make, subtract the quarter circle and everything else except for the other blue spots. We can add 4 + 7 + 3 + 2, which is 16. 13 times 16 is 208, so we can see that the big rectangle is 208 sq mm. We subtract the white and quarter circle, which gives us 208 - (6 * (7 + 3 + 2)) - (2 * 9), or 208 - 72 - 18, or 118. Then, we add the quarter circle. To get the area, we can use r² * pi. pi right now is 3.14, and the radius is 3. 3² is 9, and 3.14 * 9 is 28.26. We divide this by 4, and we get 7.065. 118 + 7.065 is equal to 125.065, so therefore the answer is 125.065. 

We have

A 2 * 7   rectangle  =  14

A 16 * 5 rectangle =  80

An 8 * 4 rectangle = 32

And a quarter-circle with a radius of 3  =  pi (3)^2 / 4  =  9/4 pi =  (9/4) 3.14 =  7.065

Add all the red numbers to get your answer

Excellent, CipherStream !!!!!

\(\alpha^2 \beta^2\)

1x^2 + 7x - 2

By Viete :

In a polynomial of the form ax^2 + bx + c, the product of the roots =   c / a

Product of the roots =  alpha * beta  =  c/a  =   -2/1 = -2       square both sides

(alpha * beta)^2  = (-2)^2

alpha^2* beta^2  = 4

Note that when x  is any negative.....the expression will evaliuate to all values on  [ -2, 0)

When x = 0   the function evaluates to  0

When x is any positive.....the function evaluates to all values on  (0, 2]

This graph is continuous for all x

So  the range is  [ -2, 2 ]

Here's a graph :

No prob....your answer was pretty close  !!!

Here :

https://web2.0calc.com/questions/in-convex-quadrilateral-abcd-ab-bc-13-cd-da-24-and_1

oh yeah I forgot the 1. Thanks, Cphill!

abs (n) < abs ( n -3) < 9       we can write

sqrt (n^2) < sqrt [ ( n -3)^2 ] < 9

Taking the first part

sqrt (n^2) < sqrt [ (n -3)^2 ]    square both sides

n^2 < (n -3)^2

n^2 < n^2 -6n + 9

0 < -6n + 9

6n < 9

n < 9/6

n < 3/2   so..... n takes on all integer values from  -infinity to 1   → (1)

Taking the second part

sqrt [ (n -3)^2 ]  < 9      square both sides

(n -3)^2  < 81

n^2 - 6n + 9 < 81

n^2 - 6n - 72 < 0

(n - 12) ( n + 6) < 0

This will be true when   n is on the imterval of (-6 ,12).....so n takes on all integer values from -5 to 11  →   (2)

However the intersection of (1) and (2) are the integers    -5, -4, -3 , -2, -1 , 0 , 1

The sum of these =   -14 

PLS HELP ASAP:

1. What is the range of the function \(G(x) = |x+1|-|x-1|~\)Express your answer in interval notation.

For your second post:

1. n must be a negative number, as all positive values of n would be greater than n-3. n must also be greater than -6, as -6-3=-9, and the absolute value of that would be nine, so 0>n>-6, so the possible integer values of n are 0, -1, -2, -3, -4, and -5. Add that up, and you get -15.

Call the time in minutes driving 40 mph =  M

Using Distance / Rate = Time......  we have in  the first case

D / 40 = M     (1)

In the second case we must convert 25 min  to hrs = 25/60hrs =  5/12 hrs....so we have that

D / 50 = M - 5/12   (2)

Sub (1) into (2)

D /50  = D/40 - 5/12   rearrange as

D/40 - D/50 =  5/12

5D / 200 - 4D /200  =  5/12

D / 200  = 5/12

D = 200 * 5/12  =  1000/12 = 250 / 3  miles =  83 1/3 miles