A set of 4 consecutive even integers has a sum of -164. What is the greatest of these integers?
The correct answer and explanation is:
To find the greatest of the four consecutive even integers whose sum is -164, let’s represent the integers algebraically.
Let the four consecutive even integers be:
- xx
- x+2x + 2
- x+4x + 4
- x+6x + 6
The sum of these integers is given as -164, so we can write the equation: x+(x+2)+(x+4)+(x+6)=−164x + (x + 2) + (x + 4) + (x + 6) = -164
Simplifying the left side: 4x+12=−1644x + 12 = -164
Now, subtract 12 from both sides to isolate the term with xx: 4x=−164−124x = -164 – 12 4x=−1764x = -176
Next, divide both sides by 4: x=−1764x = \frac{-176}{4} x=−44x = -44
Now that we know the first integer is -44, the four consecutive even integers are:
- −44-44
- −42-42
- −40-40
- −38-38
The greatest of these integers is −38-38.
Therefore, the greatest integer is −38-38.
Explanation:
The integers are consecutive even numbers, meaning each one is 2 units greater than the previous one. By representing them algebraically and setting up an equation based on their sum, we were able to solve for the first integer, which was then used to find the greatest integer. This method of setting up an equation and solving for the unknown variable is effective in solving problems involving consecutive integers.