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

Answer 1

To solve the equation

\[ \frac{3}{2}(4 - 8x) + 5x = 1 - 7x, \]

we will follow step-by-step justification using the properties of equality and simplify the equation.

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

   \[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \]

   Simplifying gives:

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

2. Combine like terms on the left side:

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

3. Add \(7x\) to both sides to isolate the constant:

   \[ 6 = 1 \]

4. Subtract 1 from both sides:

   \[ 6 - 1 = 0 \quad \Rightarrow \quad 5 = 0 \]

Since \(5 = 0\) is a contradiction (this statement is false), it indicates that there are no values of \(x\) that can satisfy the original equation.

Thus, the number of solutions to the equation is:

\[ \boxed{0} \]