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

Let's solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) step by step, using properties of equality and justifying each step.

**Step 1: Distribute the \( \frac{3}{2} \)**
\[
\frac{3}{2}(4 - 8x) = \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x = 6 - 12x
\]

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**
Using the property of equality (adding the same quantity to both sides):
\[
6 = 1
\]

**Step 4: Analyze the result**
The equation \( 6 = 1 \) is not true. This result indicates that there are no values of \( x \) that satisfy the original equation.

**Conclusion:**
The equation has no solutions. This situation is derived from the fact that we reached a false statement after simplification, which indicates that the original equation is inconsistent. Thus, the number of solutions is:
\[
\text{No solutions}
\]