To solve this equation, we can start by simplifying both sides separately.

On the left side, we have the sum of consecutive integers from x to x-100. This can be rewritten as:

x + (x-1) + (x-2) + ... + (x-100)

Using the formula for the sum of an arithmetic series, we can find the sum of this sequence:

Sum = (n/2)(first term + last term)

In this case, the first term is x and the last term is x-100. The number of terms, n, is 101 (since we are counting from x to x-100, inclusive).

Sum = (101/2)(x + (x-100))
= 101(x + x - 100)/2
= 101(2x - 100)/2
= 101(x - 50)

On the right side, we have the sum of consecutive numbers from 99 to 99.101. This can be rewritten as:

99 + 99.2 + 99.3 + ... + 99.101

Using the formula for the sum of an arithmetic series, we can find the sum of this sequence:

Sum = (n/2)(first term + last term)

In this case, the first term is 99 and the last term is 99.101. The number of terms, n, is 101.

Sum = (101/2)(99 + 99.101)
= (101/2)(99 + 9999)
= (101/2)(10098)
= 101(5049)
= 509949

Now we can set the two sums equal to each other and solve for x:

101(x - 50) = 509949

Dividing both sides by 101:

x - 50 = 5049

Adding 50 to both sides:

x = 5099

Therefore, the value of x that satisfies the equation is 5099.