First, solve for \( \left|c + 6\right| \):

\( \left|c + 6\right| \) - 4 = 3
\( \left|c + 6\right| \) = 3 + 4
\( \left|c + 6\right| \) = 7

The value inside the absolute value brackets can be either positive or negative so (c + 6) must equal + 7 or -7 for \( \left|c + 6\right| \) to equal 7:

So, c = -13 or c = 1.

To find the average of these 4 numbers add them together then divide by 4:

\( \frac{12 + 6 + 13 + 5}{4} \) = \( \frac{36}{4} \) = 9

The yearly interest charged on this loan is the annual interest rate multiplied by the amount borrowed:

interest = annual interest rate x loan amount

i = (\( \frac{6}{100} \)) x $1,500
i = 0.05 x $1,500
i = $75

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{8 x 5}{9 x 5} \) - \( \frac{2 x 3}{15 x 3} \)

\( \frac{40}{45} \) - \( \frac{6}{45} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{40 - 6}{45} \) = \( \frac{34}{45} \) = \(\frac{34}{45}\)

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 54 meters so the equation becomes: 2w + 2h = 54.

Putting these two equations together and solving for width (w):

2w + 2h = 54
w + h = \( \frac{54}{2} \)
w + h = 27
w = 27 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 27 - 2w
3w = 27
w = \( \frac{27}{3} \)
w = 9

Since h = 2w that makes h = (2 x 9) = 18 and the area = h x w = 9 x 18 = 162 m²