First, solve for \( \left|a - 4\right| \):

\( \left|a - 4\right| \) - 7 = -1
\( \left|a - 4\right| \) = -1 + 7
\( \left|a - 4\right| \) = 6

The value inside the absolute value brackets can be either positive or negative so (a - 4) must equal + 6 or -6 for \( \left|a - 4\right| \) to equal 6:

So, a = -2 or a = 10.

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

To divide terms with radicals, divide the coefficients and radicands separately:

\( \frac{10\sqrt{56}}{5\sqrt{8}} \)
\( \frac{10}{5} \) \( \sqrt{\frac{56}{8}} \)
2 \( \sqrt{7} \)

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{6!}{3!} \)
\( \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} \)
\( \frac{6 \times 5 \times 4}{1} \)
\( 6 \times 5 \times 4 \)
120

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 48 meters so the equation becomes: 2w + 2h = 48.

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

2w + 2h = 48
w + h = \( \frac{48}{2} \)
w + h = 24
w = 24 - h

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

w = 24 - 2w
3w = 24
w = \( \frac{24}{3} \)
w = 8

Since h = 2w that makes h = (2 x 8) = 16 and the area = h x w = 8 x 16 = 128 m²