To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we will use properties of equality to simplify and isolate \( x \).
### Step 1: Distribute the 32 on the left side.
\[
32(4 - 8x) = 32 \cdot 4 - 32 \cdot 8x = 128 - 256x
\]
So, the equation becomes:
\[
128 - 256x + 5x = 1 - 7x
\]
### Step 2: Combine like terms on the left side.
Combining the \( x \) terms:
\[
128 - 256x + 5x = 128 - 251x
\]
Now, the equation is:
\[
128 - 251x = 1 - 7x
\]
### Step 3: Move all \( x \) terms to one side and constant terms to the other side.
To do this, add \( 251x \) to both sides:
\[
128 = 1 - 7x + 251x
\]
Which simplifies to:
\[
128 = 1 + 244x
\]
Now, subtract \( 1 \) from both sides:
\[
128 - 1 = 244x
\]
This simplifies to:
\[
127 = 244x
\]
### Step 4: Divide both sides by \( 244 \) to solve for \( x \).
\[
x = \frac{127}{244}
\]
### Step 5: Determine the number of solutions.
Since we have found one specific solution, the number of solutions to the equation \( 32(4 - 8x) + 5x = 1 - 7x \) is **1**.
Thus, the justification of the steps used the properties of equality, specifically the distributive property, the property of combining like terms, and the additive and multiplicative properties of equality.
**Final Answer:** There is 1 solution: \( x = \frac{127}{244} \).