Four times the greater of two consecutive integers is 18 more than three times the lesser integer

The two consecutive numbers could be:

x and x+1

we know that (from what you wrote):

(x+1) * 4=18+3*x

eliminating the bracket:

4x+4=18+3*x

putting all the xs on the left side:

4x-3x=18-4

x=14

so the two consecutive numbers are 14 and 15.