We only have to translate English to mathematical symbols:
Consecutive integers means that each integer is one greater than the one to the left on the number line. Therefore, we have x, then x+1, the x+2, the x+3. These expressions represent four consecutive integers. Now let's add them. Afterall, we are working with a sum and we know that the sum is 508. The word "is" is replaced with an equal mark.
x + (x+1) + (x+2) + (x +3) = 508
Now simplify the expression on the left. We have 4 of the same variable or 4x. Count them. Our constants (called constants because 1 and 2 and 3 always occupy the same spots on the number lines - constantly. Their values are constant. Their values do not change.) have a sum of 6 (1 + 2 + 3). Now our equation is
4x + 6 = 508
Solve by cancelling the constant (6) first and balancing the equation:
4x + 6 +(-6) = 508 + (-6)
4x = 502
Now cancel the coefficient (4) and then balance the equation.
1/4(4x) = 1/4(502)
x = 125.5
x + 1 = 126.5
x + 2 = 127.5
x + 3 = 128.5
The sum of these is 508.
Were you confused with the decimal digits?