To solve the problem, we need to translate the given statement into a mathematical equation and then solve for the unknown number.
The statement given is:
"The difference of a number and 6 is the same as 5 times the sum of the number and 2."
Let's define the unknown number as [tex]\( x \)[/tex].
1. Translate the statement into an equation:
- The difference of the number and 6 can be written as [tex]\( x - 6 \)[/tex].
- The sum of the number and 2 can be written as [tex]\( x + 2 \)[/tex].
- 5 times the sum of the number and 2 is [tex]\( 5(x + 2) \)[/tex].
Therefore, the equation represents:
[tex]\[
x - 6 = 5(x + 2)
\][/tex]
2. Simplify the equation:
To solve for [tex]\( x \)[/tex], we need to simplify and isolate [tex]\( x \)[/tex]:
[tex]\[
x - 6 = 5(x + 2)
\][/tex]
3. Distribute the 5 on the right side:
[tex]\[
x - 6 = 5x + 10
\][/tex]
4. Move all terms involving [tex]\( x \)[/tex] to one side:
[tex]\[
x - 5x = 10 + 6
\][/tex]
5. Simplify the equation:
[tex]\[
-4x = 16
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{16}{-4}
\][/tex]
[tex]\[
x = -4
\][/tex]
So, the number that satisfies the given condition is [tex]\(-4\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{-4} \][/tex]
