SOLUTION: 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 a

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Question 844280: 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?
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
x is an even integer.
you are looking for the positive difference between the following:
smallest odd number greater than 7x-2
greatest odd number less than 3x+5

if x is an even number that is positive, then the positive difference will be the larger number minus the smaller number.

if x is even, then 7x-2 will always be even, so the smallest odd number greater than that will be 1 greater which will be 7x-1.

if x is even, then 3x+5 will always be odd, so the greatest number smaller than that will be 2 less which will be 3x+3.

the positive difference will be (7x-1) - (3x+3) which will be equal to 7x - 1 - 3x - 3 which will be equal to 4x - 4.

this is in the form of ax + b, where a is equal to 4 and b is equal to -4.

the equation is 4x-4.

the following table shows that this formula give you the positive difference each time.
the fourth and fifth column should be the same and they are, confirming this is true. 


x	7x-1	3x+3	(7x-1)-(3x+3)	(4x-4)
				
2	  13	  9	       4	    4
4	  27	  15	       12	    12
6	  41	  21	       20	    20
8	  55	  27	       28	    28
10	  69	  33	       36	    36
12	  83	  39	       44	    44


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