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To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\), first use the distributive property to expand the left side:
\[
32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x
\]
This simplifies to:
\[
128 - 256x + 5x = 1 - 7x
\]
Combine like terms:
\[
128 - 251x = 1 - 7x
\]
Next, isolate \(x\) by adding \(251x\) and subtracting \(1\) from both sides:
\[
128 - 1 = -7x + 251x
\]
This results in:
\[
127 = 244x
\]
Dividing both sides by \(244\) gives:
\[
x = \frac{127}{244}
\]
Since the equation leads to a single specific solution for \(x\), the number of solutions is 1.