Question

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 systematic steps and justify each action using properties of equality.

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

2. **Combine like terms** on the left side:
\[
6 - 7x = 1 - 7x
\]
(Justified as \( -12x + 5x = -7x \).)

3. **Add \( 7x \)** to both sides to eliminate \( -7x \) from both sides:
\[
6 = 1
\]
(Justified using the Addition Property of Equality.)

4. **Observe the result**: The equation \( 6 = 1 \) is a contradiction, meaning that there are no values of \( x \) that can satisfy the original equation.

Thus, the original equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) has **no solutions**.