To verify that [tex]\(-x\)[/tex] is the simplified expression of [tex]\(\frac{1}{5}(5x-20)-\frac{1}{2}(4x-8)\)[/tex], Genevieve can follow a detailed, step-by-step approach as outlined below:
### Step-by-Step Simplification Process
1. Expand the Terms Inside the Parentheses:
Let's first distribute the multiplication inside each term of the expression [tex]\(\frac{1}{5}(5x-20) - \frac{1}{2}(4x-8)\)[/tex].
Simplify [tex]\(\frac{1}{5}(5x - 20)\)[/tex]:
[tex]\[
\frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 20 = x - 4
\][/tex]
So, [tex]\(\frac{1}{5}(5x - 20) = x - 4\)[/tex].
Simplify [tex]\(\frac{1}{2}(4x - 8)\)[/tex]:
[tex]\[
\frac{1}{2} \cdot 4x - \frac{1}{2} \cdot 8 = 2x - 4
\][/tex]
So, [tex]\(\frac{1}{2}(4x - 8) = 2x - 4\)[/tex].
2. Combine the Simplified Parts:
Now, substitute the simplified terms back into the original expression:
[tex]\[
x - 4 - (2x - 4)
\][/tex]
3. Distribute and Simplify:
Next, distribute the subtraction over the second term inside the parentheses:
[tex]\[
x - 4 - 2x + 4
\][/tex]
4. Combine Like Terms:
Finally, combine the like terms to simplify the expression further:
[tex]\[
x - 2x - 4 + 4 = -x
\][/tex]
By following these steps, Genevieve can verify that the simplified expression of [tex]\(\frac{1}{5}(5x-20)-\frac{1}{2}(4x-8)\)[/tex] is indeed [tex]\(-x\)[/tex].
