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A) Quantity A is greater. B) Quantity B is greater. C) The two quantities are equal. D) The relationship cannot be determined from the information given.

A) Quantity A is greater. B) Quantity B is greater. C) The two quantities are equal. D) The relationship cannot be determined from the information given.

Great question!!!!!

Given: \(\frac{2}{d}= \frac{2-d}{d-2}\)

Cross multiply to get: 2(d - 2) = d(2 - d)

Expand: 2d - 4 = 2d - d² Subtract 2d from both sides to get: -4 = -d² Multiply both sides by -1 to get: 4 = d² So, either d = 2 or d = -2 It SEEMS that we have two solutions, but this is not the case

If we plug d = -2 into the given equation,\(\frac{2}{d}= \frac{2-d}{d-2}\), we get: \(\frac{2}{-2} = \frac{4}{-4}\) Since this equation checks out, we can conclude that d = -2 is a valid solution

HOWEVER, if we plug d = 2 into the given equation,\(\frac{2}{d}= \frac{2-d}{d-2}\), we get: \(\frac{2}{2} = \frac{0}{0}\) Since this equation does NOT check out, we can see that d = 2 is NOT a valid solution

Since there's only one possible value of d (d = -2)

I am not sure mine was the best approach but it did work.

Just by looking at the denominators, we know d will not be equal 0 or 2 because if d = (0 or 2), the denominators would be 0 which is undefined.

What happens if d is a positive number?

if d = 1 then you end up with 2 = -1 which is false. If d is greater than 2, then 2/d will ALWAYS be a positive number. The numerator (2-d) will ALWAYS be a negative number and the denominator (d-2) will ALWAYS be a positive number. Since a positive number divided by a negative number always equals a negative number, you know d cannot be a positive number since d/2 is positive and 2-d/d-2 is negative.

What if d is less than 0?

When d is a negative number, 2/d will ALWAYS be negative. The numerator (2 - d) will ALWAYS be a positive number. The denominator (d - 2) will ALWAYS be negative. A positive number divided by a negative number will always be negative.

We do not know what d will be, but at least we know it will be a negative number because that is the only way both sides of the equation will have the same sign. Since 0 is more than any negative number, we can conclude quantity b is bigger.

I hope i didn't make a mistake here. If I did, I apologize in advance.

tricky question. I thought we could not do cross multiplication due to being unsure whether denominator was going to be 0 or not. It turns out we need to assume it's not going to be 0.

A) Quantity A is greater. B) Quantity B is greater. C) The two quantities are equal. D) The relationship cannot be determined from the information given.

You can cross multiply as shown above or range fractions to get the answer straight.. \(\frac{2}{d}= \frac{2-d}{d-2}\).. \(\frac{2}{d}= \frac{-(d-2)}{(d-2)}\)=-1.. So \(\frac{2}{d}=-1\).. Only possibility is d=-2