To solve the expression \(2 - 4 + 3(\div 9 \times 12 \div (2 + 2 \times 2 \times 2 \div 2) - 1)\), we will follow the order of operations, represented by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
Let's break this down step-by-step.
Step 1: Resolve the expression inside the parentheses
Calculate \(2 + 2 \times 2 \times 2 \div 2\)
- First, calculate \(2 \times 2 \times 2 = 8\).
- Then calculate \(8 \div 2 = 4\).
- Now, add \(2 + 4 = 6\).
So the expression inside the parentheses simplifies to \(6\).
Step 2: Substitute back into the original expression
Now we replace the expression inside the parentheses:
\[ 2 - 4 + 3 \left( \div 9 \times 12 \div 6 - 1 \right) \]
Step 3: Resolve the multiplication and division in the parentheses
The expression \( \div 9 \times 12 \div 6 - 1\) is not fully clear. There might be a typographical mistake here, as divisions should be followed by numbers. However, let's assume it to be \(3(\frac{12}{9 \cdot 6} - 1)\) so:
- Start with \(12 \div 9 = \frac{4}{3}\).
- Then calculate \(\frac{4}{3} \div 6 = \frac{4}{3} \cdot \frac{1}{6} = \frac{4}{18} = \frac{2}{9}\).
Now replacing back into the expression gives us:
\[ 2 - 4 + 3\left(\frac{2}{9} - 1\right) \]
Step 4: Combine the terms inside the parentheses
Calculate \(\frac{2}{9} - 1\):
-
Convert \(1\) into a fraction with a denominator of \(9\): \[1 = \frac{9}{9}\].
-
Now, compute: \[ \frac{2}{9} - \frac{9}{9} = \frac{2 - 9}{9} = -\frac{7}{9}. \]
Step 5: Substitute back into the expression
Put the computed value back into the main expression:
\[ 2 - 4 + 3 \left(-\frac{7}{9}\right) \]
Step 6: Resolve \(3 \left(-\frac{7}{9}\right)\)
Calculate:
\[ 3 \left(-\frac{7}{9}\right) = -\frac{21}{9} = -\frac{7}{3}. \]
Step 7: Final calculation
Now we have:
\[ 2 - 4 - \frac{7}{3} \]
The \(2 - 4\) gives \(-2\), so:
\[ -2 - \frac{7}{3}. \]
To combine these, convert \(-2\) into a fraction with a denominator of \(3\): \[ -2 = -\frac{6}{3}. \]
Add:
\[ -\frac{6}{3} - \frac{7}{3} = -\frac{6 + 7}{3} = -\frac{13}{3}. \]
Final Answer
\[ \boxed{-\frac{13}{3}}. \]