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

To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) and justify the steps using properties of equality, let's follow through with the solution.

1. **Distribute** \( \frac{3}{2} \) on the left side:
\[
\frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x
\]
\[
6 - 12x + 5x = 1 - 7x
\]
(Distributive property)

2. **Combine like terms** on the left side:
\[
6 - 7x = 1 - 7x
\]
(Combining like terms)

3. **Add \( 7x \)** to both sides:
\[
6 = 1
\]
(Addition property of equality)

At this point, \( 6 = 1 \) is a contradiction and indicates that there is no solution to the equation.

4. **Conclusion:**
Since we arrived at a contradiction, this means that there are no values of \( x \) that satisfy the equation.

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