Let's analyze Audrey's work step by step to evaluate the correctness of each step in the simplification process.
The original expression is:
[tex]\[
-\frac{3}{3}\left|6x-\frac{3}{2}\right|
\][/tex]
### Step 1:
[tex]\[
\left(\frac{2}{3} \cdot 6k\right) + \left(-\frac{2}{3}, -\frac{3}{2}\right)
\][/tex]
Analysis:
- The expression [tex]\(\left(\frac{2}{3} \cdot 6k\right) \)[/tex] simplifies to [tex]\( 4k \)[/tex]. This part appears to follow the simplification correctly, where:
[tex]\[ \frac{2}{3} \cdot 6k = 4k \][/tex]
- The second part of the expression [tex]\(\left(-\frac{2}{3}, -\frac{3}{2}\right) \)[/tex] is unclear and does not make sense mathematically. The use of a comma instead of an arithmetic operation indicates an error.
- Also, the initial expression given was [tex]\(-\frac{3}{3}\left|6x - \frac{3}{2}\right|\)[/tex] which does not align with Step 1.
Conclusion: Step 1 is incorrect.
### Step 2:
[tex]\[
-4k +\left|-\frac{2}{3} \cdot -\frac{3}{2}\right|
\][/tex]
Analysis:
- On the right-hand side, [tex]\(\left|-\frac{2}{3} \cdot -\frac{3}{2}\right|\)[/tex]:
[tex]\[ -\frac{2}{3} \cdot -\frac{3}{2} = 1 \][/tex]
- Because we are dealing with an absolute value, [tex]\(|1| = 1\)[/tex].
- However, evaluating this entire step isn't directly derivable from Step 1's mistake.
Conclusion: Step 2 is incorrect.
### Step 3:
[tex]\[
-4k + 1
\][/tex]
Analysis:
- This step follows correctly from the accurate simplification from the product and the absolute value seen in the analysis of Step 2, but not from Audrey's initial Step 2 mistake.
[tex]\[ -4k + \left| 1 \right| = -4k + 1 \][/tex]
Conclusion: Step 3 is correct in its simplified forms.
### Step 4:
[tex]\[
-3t
\][/tex]
Analysis:
- Transitioning to a completely different variable [tex]\( t \)[/tex] without any prior introduction or context indicates an issue. Additionally, the term [tex]\(-3t\)[/tex] bears no logical tie to the previous simplification step.
- There is no computation step or logical reasoning that supports changing from [tex]\(-4k + 1\)[/tex] to [tex]\(-3t\)[/tex].
Conclusion: Step 4 is incorrect.
To summarize:
- Step 1 is incorrect.
- Step 2 is incorrect.
- Step 3 is correct.
- Step 4 is incorrect.
The review of Audrey's simplification process shows that the overall steps have multiple conceptual and arithmetic errors except Step 3 which is accurate on its own.
