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Given any positive even integer, \(x \), the positive difference between the smallest odd number greater than \(7x - 2\) and the largest odd number less than \(3x+5\) can be written in the form \(ax+b\). What is \(a+b\)?

 Nov 28, 2020
 #1
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3x+5 is always going to be odd given that x is even positive integer (check it and see if you can figure out why), therefore the largest odd number less than it, is (3x+5)-2 = 3x+3

 

In a similar way, 7x-2 is always going to be even, therefore the number immediately next to it the right is the smallest odd number we can find bigger than it, which we can express as (7x-2)+1 = 7x-1

 

Now we can just find the difference between the two expressions we figued out. 

7x -1 - (3x+3) = 7x -1 -3x -3 = 4x -4

 Nov 28, 2020
 #2
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That makes sense, thank you!

Guest Nov 28, 2020

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