Justify your steps using the properties of equality and determine the number of solutions to the equation 3/2 (4-8x)+5x=1-7x

To solve the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x \) and justify each step using the properties of equality, we will follow these steps:

### Step 1: Distribute on the left-hand side
We start with the equation:
\[
\frac{3}{2} (4 - 8x) + 5x = 1 - 7x
\]
Distributing \( \frac{3}{2} \) gives:
\[
\frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x
\]
Calculating the distribution:
\[
6 - 12x + 5x = 1 - 7x
\]
Thus, we simplify the left-hand side:
\[
6 - 7x = 1 - 7x
\]

### Step 2: Add \( 7x \) to both sides
To isolate the constant term on the left, we add \( 7x \) to both sides:
\[
6 - 7x + 7x = 1 - 7x + 7x
\]
This simplifies to:
\[
6 = 1
\]

### Step 3: Analyze the result
The equation \( 6 = 1 \) is a contradiction. Since this statement is not true, it indicates that there is no value of \( x \) that can satisfy the original equation.

### Conclusion
Thus, there are **no solutions** to the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x \).

Using properties of equality:
- We used the **distributive property** to expand \( \frac{3}{2}(4 - 8x) \).
- We used the **addition property of equality** to add \( 7x \) to both sides of the equation.
- The contradiction \( 6 = 1 \) indicates there are no solutions.