Suppose that the weight of a single marble is represented by the expression [tex]\frac{x+1}{x-8}[/tex], where [tex]x[/tex] is a nonzero whole number. If a bag contains [tex]\frac{2x-16}{x-1}[/tex] marbles, which expression represents the weight of all the marbles in the bag?

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]



Answer :

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]