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 \(\frac{3}{2}(4 - 8x) + 5x = 1 - 7x\), we will apply the properties of equality step by step:

**Step 1: Distribute \(\frac{3}{2}\) into \(4 - 8x\)**

\[
\frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x = \frac{12}{2} - \frac{24}{2}x = 6 - 12x
\]

So, the equation becomes:

\[
6 - 12x + 5x = 1 - 7x
\]

**Step 2: Combine like terms on the left side**

Combine \(-12x\) and \(5x\):

\[
6 - 7x = 1 - 7x
\]

**Step 3: Add \(7x\) to both sides**

This helps to eliminate \(-7x\) from both sides:

\[
6 - 7x + 7x = 1 - 7x + 7x
\]

This simplifies to:

\[
6 = 1
\]

**Step 4: Analyze the resulting equation**

Since we have the statement \(6 = 1\), which is a contradiction (this statement is false), this indicates that there are no possible values of \(x\) that can satisfy the original equation.

**Conclusion**:

Thus, the number of solutions to the equation \(\frac{3}{2}(4 - 8x) + 5x = 1 - 7x\) is \(\boxed{0}\).