Let's solve this problem step by step:
1. Define the integers: Let [tex]\( x \)[/tex] be the smallest even integer. Then, the next even integer would be [tex]\( x + 2 \)[/tex], and the largest even integer would be [tex]\( x + 4 \)[/tex].
2. Write the equation: We know that the sum of these three integers equals -78. Therefore, we can write the equation:
[tex]\[
x + (x + 2) + (x + 4) = -78
\][/tex]
3. Simplify the equation: Combine like terms:
[tex]\[
x + x + 2 + x + 4 = -78
\][/tex]
[tex]\[
3x + 6 = -78
\][/tex]
4. Solve the equation: To isolate [tex]\( x \)[/tex]:
[tex]\[
3x + 6 = -78
\][/tex]
Subtract 6 from both sides:
[tex]\[
3x = -78 - 6
\][/tex]
[tex]\[
3x = -84
\][/tex]
Divide both sides by 3:
[tex]\[
x = -28
\][/tex]
5. Find the integers:
- The smallest even integer is [tex]\( x = -28 \)[/tex].
- The next even integer is [tex]\( x + 2 = -28 + 2 = -26 \)[/tex].
- The largest even integer is [tex]\( x + 4 = -28 + 4 = -24 \)[/tex].
6. Check the solution: Sum the integers to ensure they equal -78.
[tex]\[
-28 + (-26) + (-24) = -28 - 26 - 24 = -78
\][/tex]
Hence, the solution to the problem is that the smallest even integer is [tex]\(-28\)[/tex], the next even integer is [tex]\(-26\)[/tex], and the largest even integer is [tex]\(-24\)[/tex].
