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 the steps systematically, justifying each step using properties of equality.

### Step 1: Distribute \(\frac{3}{2}\) on the left side.
We start by distributing \( \frac{3}{2} \) to both terms inside the parentheses:
\[
\frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x
\]
Calculating each term:
\[
6 - 12x + 5x = 1 - 7x
\]
(Justification: Distributive Property)

### Step 2: Combine like terms on the left side.
Now, we combine the \(x\) terms:
\[
6 - 12x + 5x = 6 - 7x
\]
This simplifies to:
\[
6 - 7x = 1 - 7x
\]
(Justification: Combining Like Terms)

### Step 3: Add \(7x\) to both sides.
To isolate the constant on the left side, we will add \(7x\) to both sides of the equation:
\[
6 - 7x + 7x = 1 - 7x + 7x
\]
This gives:
\[
6 = 1
\]
(Justification: Addition Property of Equality)

### Step 4: Analyze the result.
The equation \(6 = 1\) is a contradiction; it is not true for any value of \(x\).

### Conclusion
Since we reached a contradiction, there are **no solutions** to the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x\).

In summary, we determined that the number of solutions to the equation is **zero**, justified by applying the properties of equality, and identifying that a false statement results from our manipulation of the equation.