Let's solve the problem using the setup provided.
We are given three consecutive even integers:
- 1st integer: \( x \)
- 2nd integer: \( x + 2 \)
- 3rd integer: \( x + 4 \)
The equation based on the sum of these integers is:
\[ x + (x + 2) + (x + 4) = 120 \]
Now, we can simplify the left side of the equation:
\[ x + x + 2 + x + 4 = 120 \]
Combining like terms:
\[ 3x + 6 = 120 \]
Next, we'll isolate \( x \) by first subtracting \( 6 \) from both sides:
\[ 3x = 120 - 6 \]
\[ 3x = 114 \]
Now, divide both sides by \( 3 \):
\[ x = \frac{114}{3} \]
\[ x = 38 \]
Now we can find the three consecutive even integers:
- 1st integer: \( x = 38 \)
- 2nd integer: \( x + 2 = 38 + 2 = 40 \)
- 3rd integer: \( x + 4 = 38 + 4 = 42 \)
Thus, the three consecutive even integers are \( 38, 40, \) and \( 42 \).
Let's verify the solution by calculating their sum:
\[ 38 + 40 + 42 = 120 \]
Since the sum is correct, the solution is confirmed. Therefore, the three consecutive even integers are 38, 40, and 42.