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.