To justify that the expression [tex]\(\frac{1}{6}(6x + 12) - \frac{1}{2}(4x + 2)\)[/tex] is equivalent to [tex]\(-x + 1\)[/tex], Darren and Quincy can substitute [tex]\(x = 6\)[/tex] into both expressions and evaluate them. Let's go through the detailed steps for each expression.
### Step-by-Step Solution
#### Evaluate [tex]\(\frac{1}{6}(6 x + 12) - \frac{1}{2}(4 x + 2)\)[/tex]:
1. Substitute [tex]\(x = 6\)[/tex] into the expression:
[tex]\[
\frac{1}{6}(6(6) + 12) - \frac{1}{2}(4(6) + 2)
\][/tex]
2. Simplify the terms inside the parentheses:
[tex]\[
(6(6) + 12) = (36 + 12) = 48
\][/tex]
[tex]\[
(4(6) + 2) = (24 + 2) = 26
\][/tex]
3. Continue with the expression:
[tex]\[
\frac{1}{6}(48) - \frac{1}{2}(26)
\][/tex]
4. Simplify each fraction:
[tex]\[
\frac{1}{6}(48) = 8
\][/tex]
[tex]\[
\frac{1}{2}(26) = 13
\][/tex]
5. Subtract the two results:
[tex]\[
8 - 13 = -5
\][/tex]
So, the value of [tex]\(\frac{1}{6}(6x + 12) - \frac{1}{2}(4x + 2)\)[/tex] when [tex]\(x = 6\)[/tex] is [tex]\(-5\)[/tex].
#### Evaluate [tex]\(-x + 1\)[/tex]:
1. Substitute [tex]\(x = 6\)[/tex] into the expression:
[tex]\[
-6 + 1
\][/tex]
2. Simplify:
[tex]\[
-6 + 1 = -5
\][/tex]
So, the value of [tex]\(-x + 1\)[/tex] when [tex]\(x = 6\)[/tex] is also [tex]\(-5\)[/tex].
#### Conclusion:
Since both expressions result in [tex]\(-5\)[/tex] when [tex]\(x = 6\)[/tex], we can conclude that [tex]\(\frac{1}{6}(6x + 12) - \frac{1}{2}(4x + 2)\)[/tex] is equivalent to [tex]\(-x + 1\)[/tex].
