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 determine the number of solutions, we will follow a series of steps while justifying each step using the properties of equality.

### Step 1: Distribute the \( \frac{3}{2} \) on the left side

Starting with the equation:
\[
\frac{3}{2}(4 - 8x) + 5x = 1 - 7x
\]

Using the distributive property:
\[
\frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x
\]
Calculating the multiplication:
\[
6 - 12x + 5x = 1 - 7x
\]

### Step 2: Combine like terms

Now we simplify the left side:
\[
6 - 12x + 5x = 6 - 7x
\]
So the equation now looks like:
\[
6 - 7x = 1 - 7x
\]

### Step 3: Isolate the variable

We notice that both sides contain \(-7x\). Let's add \(7x\) to both sides to eliminate it:
\[
6 - 7x + 7x = 1 - 7x + 7x
\]
This simplifies to:
\[
6 = 1
\]

### Step 4: Analyze the resulting statement

The equation \(6 = 1\) is a false statement. This indicates that there are **no values of \(x\)** that can satisfy the original equation.

### Conclusion

Hence, the number of solutions to the equation \( \frac{3}{2}(4-8x) + 5x = 1 - 7x \) is:
\[
\text{No solutions}
\]