Determinate the integers x for which the fraction (3x+2)/(4x+5) can be simplified with 7.

If the fraction can be simplified by 7, taht means that
the numerator and denominator are multiple of 7:

3x + 2 =
7n

We'll isolate x to the left side and we'll keep in mind
that we'll ahve to get an integer value for x:

3x = 7n -
2

x = (7n-2)/3

For x to be
integer, than 7n - 2 has to be divided by 3:

If n = 2
=> 7*2 - 2 = 14 - 2 = 12

12 is divided by 3
=> x = 4 for n = 2

If n = 5 => x =
11

If n = 8 => x =
18...

Also, the denominator has to be multiple of
7:

4x + 5 = 7k => x =
(7k-5)/4

We'll put x = 4 => 16 = 7k - 5 => 7k
= 21 => k = 3

We'll put x = 11 => 44 = 7k-5
=> 49 = 7k => k = 7

We'll put x = 18
=> 72= 7k-5 => 77 = 7k => k =
11

Therefore, the values of x :{4 ; 11 ; 18
....} are the terms of an arithmetical progression, whose common difference is d = 7 and
the first term is 4.