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Let's go through the step-by-step process for matching each mathematical expression with its simplified form based on the results.
Consider the expressions and their numbering:
1. [tex]\(\frac{3(3 x-1)}{-3 x+2}\)[/tex]
2. [tex]\(\frac{-3 x+2}{3(3 x-1)}\)[/tex]
3. [tex]\(\frac{12}{(3 x-1)(-3 x+2)}\)[/tex]
4. [tex]\(\frac{2(12 x+1)}{(3 x-1)(-3 x+2)}\)[/tex]
5. [tex]\(\frac{2(6 x-1)}{(3 x-1)(-3 x+2)}\)[/tex]
6. [tex]\(\frac{-2(12 x-5)}{(3 x-1)(-3 x+2)}\)[/tex]
7. [tex]\(P(x) \cdot Q(x)\)[/tex]
8. [tex]\(P(x) \div Q(x)\)[/tex]
Here are the simplified forms results from calculations:
Simplified form 1: [tex]\(-6.857142857142857\)[/tex] (Expression: [tex]\(\frac{3(3 x-1)}{-3 x+2}\)[/tex])
Simplified form 2: [tex]\(-6.222222222222221\)[/tex] (Expression: [tex]\(\frac{-3 x+2}{3(3 x-1)}\)[/tex])
Simplified form 3: [tex]\(-0.9642857142857142\)[/tex] (Expression: [tex]\(\frac{12}{(3 x-1)(-3 x+2)}\)[/tex])
Simplified form 4: [tex]\(-0.970982142857143\)[/tex] (Expression: [tex]\(\frac{2(12 x+1)}{(3 x-1)(-3 x+2)}\)[/tex])
Simplified form 5: [tex]\(-0.49553571428571436\)[/tex] (Expression: [tex]\(\frac{2(6 x-1)}{(3 x-1)(-3 x+2)}\)[/tex])
Simplified form 6: [tex]\(1.2410714285714286\)[/tex] (Expression: [tex]\(\frac{-2(12 x-5)}{(3 x-1)(-3 x+2)}\)[/tex])
Now for [tex]\(P(x) \cdot Q(x)\)[/tex] and [tex]\(P(x) \div Q(x)\)[/tex], we have:
- [tex]\(P(x) \cdot Q(x) = -1.9285714285714284\)[/tex]
- [tex]\(P(x) \div Q(x) = -0.2916666666666667\)[/tex]
Matching expressions with simplified forms:
[tex]\[ \frac{3(3 x-1)}{-3 x+2} \quad \longrightarrow \quad -6.857142857142857 \][/tex]
[tex]\[ \frac{-3 x+2}{3(3 x-1)} \quad \longrightarrow \quad -6.222222222222221 \][/tex]
[tex]\[ \frac{12}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad -0.9642857142857142 \][/tex]
[tex]\[ \frac{2(12 x+1)}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad -0.970982142857143 \][/tex]
[tex]\[ \frac{2(6 x-1)}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad -0.49553571428571436 \][/tex]
[tex]\[ \frac{-2(12 x-5)}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad 1.2410714285714286 \][/tex]
[tex]\[ P(x) \cdot Q(x) \quad \longrightarrow \quad -1.9285714285714284 \][/tex]
[tex]\[ P(x) \div Q(x) \quad \longrightarrow \quad -0.2916666666666667 \][/tex]
Therefore, the correct matches are:
[tex]\[ \frac{3(3 x-1)}{-3 x+2} \quad \text{= } \boxed{-6.857142857142857} \][/tex]
[tex]\[ \frac{-3 x+2}{3(3 x-1)} \quad \text{= } \boxed{-6.222222222222221} \][/tex]
[tex]\[ \frac{12}{(3 x-1)(-3 x+2)} \quad \text{= } \boxed{-0.9642857142857142} \][/tex]
[tex]\[ \frac{2(12 x+1)}{(3 x-1)(-3 x+2)} \quad \boxed{-0.970982142857143} \][/tex]
[tex]\[ \frac{2(6 x-1)}{(3 x-1)(-3 x+2)} \quad \boxed{-0.49553571428571436} \][/tex]
[tex]\[ \frac{-2(12 x-5)}{(3 x-1)(-3 x+2)} \quad \boxed{1.2410714285714286} \][/tex]
[tex]\[ P(x) \cdot Q(x) \quad \boxed{-1.9285714285714284} \][/tex]
[tex]\[ P(x) \div Q(x) \quad \boxed{-0.2916666666666667} \][/tex]
Consider the expressions and their numbering:
1. [tex]\(\frac{3(3 x-1)}{-3 x+2}\)[/tex]
2. [tex]\(\frac{-3 x+2}{3(3 x-1)}\)[/tex]
3. [tex]\(\frac{12}{(3 x-1)(-3 x+2)}\)[/tex]
4. [tex]\(\frac{2(12 x+1)}{(3 x-1)(-3 x+2)}\)[/tex]
5. [tex]\(\frac{2(6 x-1)}{(3 x-1)(-3 x+2)}\)[/tex]
6. [tex]\(\frac{-2(12 x-5)}{(3 x-1)(-3 x+2)}\)[/tex]
7. [tex]\(P(x) \cdot Q(x)\)[/tex]
8. [tex]\(P(x) \div Q(x)\)[/tex]
Here are the simplified forms results from calculations:
Simplified form 1: [tex]\(-6.857142857142857\)[/tex] (Expression: [tex]\(\frac{3(3 x-1)}{-3 x+2}\)[/tex])
Simplified form 2: [tex]\(-6.222222222222221\)[/tex] (Expression: [tex]\(\frac{-3 x+2}{3(3 x-1)}\)[/tex])
Simplified form 3: [tex]\(-0.9642857142857142\)[/tex] (Expression: [tex]\(\frac{12}{(3 x-1)(-3 x+2)}\)[/tex])
Simplified form 4: [tex]\(-0.970982142857143\)[/tex] (Expression: [tex]\(\frac{2(12 x+1)}{(3 x-1)(-3 x+2)}\)[/tex])
Simplified form 5: [tex]\(-0.49553571428571436\)[/tex] (Expression: [tex]\(\frac{2(6 x-1)}{(3 x-1)(-3 x+2)}\)[/tex])
Simplified form 6: [tex]\(1.2410714285714286\)[/tex] (Expression: [tex]\(\frac{-2(12 x-5)}{(3 x-1)(-3 x+2)}\)[/tex])
Now for [tex]\(P(x) \cdot Q(x)\)[/tex] and [tex]\(P(x) \div Q(x)\)[/tex], we have:
- [tex]\(P(x) \cdot Q(x) = -1.9285714285714284\)[/tex]
- [tex]\(P(x) \div Q(x) = -0.2916666666666667\)[/tex]
Matching expressions with simplified forms:
[tex]\[ \frac{3(3 x-1)}{-3 x+2} \quad \longrightarrow \quad -6.857142857142857 \][/tex]
[tex]\[ \frac{-3 x+2}{3(3 x-1)} \quad \longrightarrow \quad -6.222222222222221 \][/tex]
[tex]\[ \frac{12}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad -0.9642857142857142 \][/tex]
[tex]\[ \frac{2(12 x+1)}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad -0.970982142857143 \][/tex]
[tex]\[ \frac{2(6 x-1)}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad -0.49553571428571436 \][/tex]
[tex]\[ \frac{-2(12 x-5)}{(3 x-1)(-3 x+2)} \quad \longrightarrow \quad 1.2410714285714286 \][/tex]
[tex]\[ P(x) \cdot Q(x) \quad \longrightarrow \quad -1.9285714285714284 \][/tex]
[tex]\[ P(x) \div Q(x) \quad \longrightarrow \quad -0.2916666666666667 \][/tex]
Therefore, the correct matches are:
[tex]\[ \frac{3(3 x-1)}{-3 x+2} \quad \text{= } \boxed{-6.857142857142857} \][/tex]
[tex]\[ \frac{-3 x+2}{3(3 x-1)} \quad \text{= } \boxed{-6.222222222222221} \][/tex]
[tex]\[ \frac{12}{(3 x-1)(-3 x+2)} \quad \text{= } \boxed{-0.9642857142857142} \][/tex]
[tex]\[ \frac{2(12 x+1)}{(3 x-1)(-3 x+2)} \quad \boxed{-0.970982142857143} \][/tex]
[tex]\[ \frac{2(6 x-1)}{(3 x-1)(-3 x+2)} \quad \boxed{-0.49553571428571436} \][/tex]
[tex]\[ \frac{-2(12 x-5)}{(3 x-1)(-3 x+2)} \quad \boxed{1.2410714285714286} \][/tex]
[tex]\[ P(x) \cdot Q(x) \quad \boxed{-1.9285714285714284} \][/tex]
[tex]\[ P(x) \div Q(x) \quad \boxed{-0.2916666666666667} \][/tex]
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