To evaluate the expression \( |4 - 6| - |9 + 3| + 2 |1 - 6| \), we will calculate each absolute value step by step.
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Calculate \( |4 - 6| \): \[ 4 - 6 = -2 \quad \Rightarrow \quad |4 - 6| = |-2| = 2 \]
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Calculate \( |9 + 3| \): \[ 9 + 3 = 12 \quad \Rightarrow \quad |9 + 3| = |12| = 12 \]
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Calculate \( |1 - 6| \): \[ 1 - 6 = -5 \quad \Rightarrow \quad |1 - 6| = |-5| = 5 \]
Now substituting these values back into the expression:
\[ |4 - 6| - |9 + 3| + 2 |1 - 6| = 2 - 12 + 2 \times 5 \]
Next, compute \( 2 \times 5 \):
\[ 2 \times 5 = 10 \]
Now substitute that back into the expression:
\[ 2 - 12 + 10 \]
Now, perform the operations step by step:
\[ 2 - 12 = -10 \]
Then add \( 10 \):
\[ -10 + 10 = 0 \]
Thus, the value of the expression is
\[ \boxed{0} \]