To determine the total weight of all the marbles in the bag, we need to multiply the weight of a single marble by the total number of marbles:
1. Weight of a single marble is given by the expression:
[tex]\[
\frac{x + 1}{x - 8}
\][/tex]
2. Number of marbles in the bag is given by the expression:
[tex]\[
\frac{2x - 16}{x - 1}
\][/tex]
We need to find the expression that represents the total weight of all the marbles in the bag. This can be done by multiplying these two expressions:
3. Multiply the expressions:
[tex]\[
\left(\frac{x + 1}{x - 8}\right) \times \left(\frac{2x - 16}{x - 1}\right)
\][/tex]
4. Simplify the expression step-by-step:
[tex]\[
\frac{(x + 1) \cdot (2x - 16)}{(x - 8) \cdot (x - 1)}
\][/tex]
Let's simplify the numerator and the denominator individually.
Numerator:
[tex]\[
(x + 1)(2x - 16) = x \cdot 2x + x \cdot (-16) + 1 \cdot 2x + 1 \cdot (-16)
= 2x^2 - 16x + 2x - 16
= 2x^2 - 14x - 16
\][/tex]
Denominator:
[tex]\[
(x - 8)(x - 1) = x \cdot x + x \cdot (-1) - 8 \cdot x - 8 \cdot (-1)
= x^2 - x - 8x + 8
= x^2 - 9x + 8
\][/tex]
So we have:
[tex]\[
\frac{(2x^2 - 14x - 16)}{(x^2 - 9x + 8)}
\][/tex]
5. Further simplify this fraction.
But we should check directly the forms given in the options.
We are given the following possible options:
- A. [tex]\( \frac{2x + 2}{x - 1} \)[/tex]
- B. [tex]\( \frac{2x^2 - 2}{x^2 + 1} \)[/tex]
- C. [tex]\( \frac{x - 8}{x - 1} \)[/tex]
- D. [tex]\( \frac{2x^2 - 16}{x^2 + 8} \)[/tex]
Looking at our simplified result and comparing with the options, it is clear that the expression simplifies down straightforwardly to:
[tex]\[
2 \cdot \frac{x + 1}{x - 1}
\][/tex]
Hence, the correct answer is:
- A. [tex]\( \frac{2(x + 1)}{x - 1} \)[/tex]
The expression that represents the weight of all the marbles in the bag is therefore:
[tex]\[
\boxed{\frac{2(x + 1)}{x - 1}}
\][/tex]
