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 \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), let's start by simplifying and justifying the steps using properties of equality.

### Step 1: Distribute \( \frac{3}{2} \)
Using the distributive property:
\[
\frac{3}{2}(4) + \frac{3}{2}(-8x) + 5x = 1 - 7x
\]
This simplifies to:
\[
6 - 12x + 5x = 1 - 7x
\]

### Step 2: Combine like terms on the left side
Combine the \( x \) terms:
\[
6 - 12x + 5x = 6 - 7x
\]
So we have:
\[
6 - 7x = 1 - 7x
\]

### Step 3: Isolate the constant terms
To isolate the constant, we can subtract \( 1 \) from both sides:
\[
6 - 1 - 7x = -7x
\]
This simplifies to:
\[
5 - 7x = -7x
\]

### Step 4: Add \( 7x \) to both sides
Using the addition property of equality, we add \( 7x \) to both sides:
\[
5 - 7x + 7x = -7x + 7x
\]
This simplifies to:
\[
5 = 0
\]

### Step 5: Interpretation
The statement \( 5 = 0 \) is a contradiction. This indicates that there are no values of \( x \) that would satisfy the original equation.

### Conclusion
The original equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) has **no solutions**. This conclusion is based on the fact that we arrived at a contradictory statement, indicating that the necessary condition for any \( x \) is impossible.