First, solve for \( \left|x - 8\right| \):

\( \left|x - 8\right| \) + 1 = -4
\( \left|x - 8\right| \) = -4 - 1
\( \left|x - 8\right| \) = -5

The value inside the absolute value brackets can be either positive or negative so (x - 8) must equal - 5 or --5 for \( \left|x - 8\right| \) to equal -5:

So, x = 13 or x = 3.

To divide fractions, invert the second fraction and then multiply:

\( \frac{3}{6} \) ÷ \( \frac{4}{9} \) = \( \frac{3}{6} \) x \( \frac{9}{4} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{3}{6} \) x \( \frac{9}{4} \) = \( \frac{3 x 9}{6 x 4} \) = \( \frac{27}{24} \) = 1\(\frac{1}{8}\)

The amount of flour you need is (2\(\frac{3}{4}\) - 1\(\frac{1}{8}\)) cups. Rewrite the quantities so they share a common denominator and subtract:

(\( \frac{22}{8} \) - \( \frac{9}{8} \)) cups
\( \frac{13}{8} \) cups
1\(\frac{5}{8}\) cups

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:

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

A ratio of 5:1 means that there are 5 home fans for every one visiting fan. So, of every 6 fans, 5 are home fans and \( \frac{5}{6} \) of every fan in the stadium is a home fan:

37,000 fans x \( \frac{5}{6} \) = \( \frac{185000}{6} \) = 30,833 fans.