To solve the problem of finding three consecutive integers whose sum is -14, we start by defining the integers and forming an equation.
1. Define the integers:
- Let [tex]\( x \)[/tex] represent the smallest of the three consecutive integers.
- Therefore, the next two integers would be [tex]\( x+1 \)[/tex] and [tex]\( x+2 \)[/tex].
2. Form the equation based on the sum:
- The sum of these three consecutive integers is given by:
[tex]\[
x + (x + 1) + (x + 2) = -14
\][/tex]
3. Simplify the equation:
- Combine like terms:
[tex]\[
x + x + 1 + x + 2 = -14
\][/tex]
[tex]\[
3x + 3 = -14
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- First, isolate the term with [tex]\( x \)[/tex] by subtracting 3 from both sides:
[tex]\[
3x + 3 - 3 = -14 - 3
\][/tex]
[tex]\[
3x = -17
\][/tex]
- Next, solve for [tex]\( x \)[/tex] by dividing both sides by 3:
[tex]\[
x = \frac{-17}{3}
\][/tex]
5. Interpret the result:
- The value [tex]\( x = \frac{-17}{3} \)[/tex] is not an integer. Since [tex]\( x \)[/tex] must represent an integer (the smallest one in this context), it indicates there is no integer solution for this equation.
- The problem necessitates that [tex]\( x \)[/tex] be an integer, hence the sum of three consecutive integers cannot equal -14 under these conditions.
Therefore:
[tex]\[
\boxed{\text{The sum of three consecutive integers cannot be -14 with integer values.}}
\][/tex]
