Every day, dave eats either a sandwich or pizza for lunch. over 42 days, dave had pizza 3 times for every 4 times he had a sandwich. over the next x days, he had pizza 3 times for every 2 times he had a sandwich. if at the end of this entire period he had pizza as many times as he had a sandwich, what is the value of x?

There are at least a couple of ways to think about this.
1) The fraction of the first 42 days that Dave ate pizza was 3/(3+4) = 3/7, so Dave ate (3/7)·42 = 18 pizzas during that time. The fraction of the next x days that Dave at pizza was 3/(3+2) = 3/5. We want the total number of pizzas to be half the total number of days.
  18 + (3/5)x = (1/2)(42 + x)
  x(3/5 -1/2) = (1/2)·42 - 18 . . . . . subtract 1/2x + 18
  x = 3/(1/10) . . . . . . . . . . . . . . . . .divide by the coeffient of x
  x = 30

2) Dave has 1 less pizza than sandwich in each period of 7 days, so over 42 days he has 6 fewer pizzas than sandwiches. Dave has 1 more pizza than sandwich in each period of 5 days for the next x days. In order for that to be 6 more pizzas than sandwiches (bringing the difference to zero), the value of x must be
  x = 6·5 = 30