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:

41,000 fans x \( \frac{2}{3} \) = \( \frac{82000}{3} \) = 27,333 fans.

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.

Use PEMDAS (Parentheses, Exponents, Multipy/Divide, Add/Subtract):

3 + (2 + 5) ÷ 3 x 2 - 2²
P: 3 + (7) ÷ 3 x 2 - 2²
E: 3 + 7 ÷ 3 x 2 - 4
MD: 3 + \( \frac{7}{3} \) x 2 - 4
MD: 3 + \( \frac{14}{3} \) - 4
AS: \( \frac{9}{3} \) + \( \frac{14}{3} \) - 4
AS: \( \frac{23}{3} \) - 4
AS: \( \frac{23 - 12}{3} \)
\( \frac{11}{3} \)
3\(\frac{2}{3}\)

First, solve for \( \left|z + 2\right| \):

\( \left|z + 2\right| \) + 3 = 8
\( \left|z + 2\right| \) = 8 - 3
\( \left|z + 2\right| \) = 5

The value inside the absolute value brackets can be either positive or negative so (z + 2) must equal + 5 or -5 for \( \left|z + 2\right| \) to equal 5:

So, z = -7 or z = 3.

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