+0  
 
0
63
1
avatar

What is the sum of all integer solutions to \($|n|<|n-3|<9$\)?

 Aug 19, 2022
 #1
avatar+124596 
+1

Note that  abs (n)   = sqrt (n^2)

And  abs ( n -3)  =  sqrt [ (n -3)^2 ] 

And  9 =  sqrt (9^2)

So

 

sqrt (n^2)  < sqrt [ (n -3)^2 ]

n^2  <  (n - 3)^2

n^2  < n^2 - 6n + 9

6n < 9

n <  9/6

n <  3/2

 

And

sqrt [ (n -3)^2 ]  < sqrt (9^2)

(n - 3)^2 < 9^2

n^2 - 6n + 9  < 81

n^2 - 6n - 72 < 0

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

This is true  if  n is on the interval

( -6 , 12)

 

Choosing  the least restrictive interval we have that

n = (-6, 3/2)

 

The integers in this interval  =  -5 , -4 ,-3, -2 , -1, 0 , 1

 

Their sum  =  -14

 

 

cool cool cool

 Aug 19, 2022

11 Online Users