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: \( x = \frac{127}{244} \).