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

\( \left|x + 6\right| \) + 9 = 7
\( \left|x + 6\right| \) = 7 - 9
\( \left|x + 6\right| \) = -2

The value inside the absolute value brackets can be either positive or negative so (x + 6) must equal - 2 or --2 for \( \left|x + 6\right| \) to equal -2:

So, x = -4 or x = -8.

To raise a term with an exponent to another exponent, retain the base and multiply the exponents:

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

\( \frac{2}{6} \) ÷ \( \frac{3}{6} \) = \( \frac{2}{6} \) x \( \frac{6}{3} \)

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

\( \frac{2}{6} \) x \( \frac{6}{3} \) = \( \frac{2 x 6}{6 x 3} \) = \( \frac{12}{18} \) = \(\frac{2}{3}\)

To simplify this fraction, first find the greatest common factor between them. The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 44 are [1, 2, 4, 11, 22, 44]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{40}{44} \) = \( \frac{\frac{40}{4}}{\frac{44}{4}} \) = \( \frac{10}{11} \)

A ratio of 2:1 means that there are 2 home fans for every one visiting fan. So, of every 3 fans, 2 are home fans and \( \frac{2}{3} \) of every fan in the stadium is a home fan:

47,000 fans x \( \frac{2}{3} \) = \( \frac{94000}{3} \) = 31,333 fans.