d can't be possible a, c, e are the only right Choices for this question

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Solution:

Let's explore each option on an individual basis here:

Option A)
c < m < 0

Let's take the point G as (c, d) implies (-5, 0) and the point J as (m, n) implies (-4, 2), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{\Delta y}{\Delta x}\)

Slope of line \(\overline{GJ}\) = \(\frac{{2-0}}{{-4-(-5)}}\) = 2

Therefore, this option may be true.

Option B)
d > n > 0

Let's take the point G as (c, d) implies (-5, 3) and the point J as (m, n) implies (-4, 1), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)

Slope of line \(\overline{GJ}\) = \(\frac{{3 - 1}}{{-5 - (-4)}}\) = -2

Similarly, if we take the point G as (c, d) implies (5, 10) and the point J as (m, n) implies (7, 8), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)

Slope of line \(\overline{GJ}\) = \(\frac{{10 - 8}}{{5 - 7}}\)= -1

That's why, this option can not be true.

Option C)
n > d > 0

The points G and J can fall in the 1st or 2nd quadrant. Let's take the point G as (c, d) implies (-6, 7) and the point J as (m, n) implies (-4, 9), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)

Slope of line \(\overline{GJ}\) = \(\frac{{9 - 7}}{{-4 - (-6)}}\) = 2

Therefore, this option can be true.

Option D)
n = 0

We can assume the point G as (c, d) implies (-6, 6) and the point J as (m, n) implies (-4, 0), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)

Slope of line \(\overline{GJ}\) = \(\frac{{6 - 0}}{{-6 - (-4)}}\) = -3

Similarly, if we take the point G as (c, d) implies (-6, -6) and the point J as (m, n) implies (-4, 0), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)

Slope of line \(\overline{GJ}\) = \(\frac{{-6 - 0}}{{-6 - (-4)}}\)= 3
Therefore, this option can be true.

Option E)
d < 0 < n

Let's take the point G as (c, d) implies (-5, -2) and the point J as (m, n) implies (-3, 2), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)

Slope of line \(\overline{GJ}\) = \(\frac{{2 - (-2)}}{{-3 - (-5)}}\) = 2

That's why, this option may be true as well.

Hence, the correct answers are Option A), C), D) and E).
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