Answered

Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.

Consider the functions given below.
[tex]\[
\begin{array}{l}
P(x) = \frac{2}{3x-1} \\
Q(x) = \frac{6}{3x+2}
\end{array}
\][/tex]

Match each expression with its simplified form.

[tex]\[
\frac{3(3x-1)}{-3x+2}
\][/tex]
[tex]\[
\frac{-3x+2}{3(3x-1)}
\][/tex]

[tex]\[
\begin{array}{l}
\frac{12}{(3x-1)(-3x+2)} \\
\frac{2(12x+1)}{(3x-1)(-3x+2)} \\
\frac{2(6x-1)}{(3x-1)(-3x+2)} \\
\end{array}
\][/tex]

[tex]\[
\square
\][/tex]

[tex]\[
\frac{-2(12x-5)}{(3x-1)(-3x+2)}
\][/tex]

[tex]\[
P(x) \cdot Q(x)
\][/tex]

[tex]\[
\square
\][/tex]

[tex]\[
P(x) \div Q(x)
\][/tex]

[tex]\[
\square
\][/tex]



Answer :

GinnyAnswer
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]

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