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Bot GPT 4-o mini To solve the equation
\[
\frac{3}{2}(4 - 8x) + 5x = 1 - 7x,
\]
we will apply the properties of equality step by step.
**Step 1: Distribute \(\frac{3}{2}\)**
Using the distributive property:
\[
\frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x.
\]
This simplifies to:
\[
6 - 12x + 5x = 1 - 7x.
\]
**Step 2: Combine like terms on the left side**
Now we combine the \(x\) terms:
\[
6 - 12x + 5x = 6 - 7x.
\]
So, we rewrite the equation as:
\[
6 - 7x = 1 - 7x.
\]
**Step 3: Add \(7x\) to both sides**
Next, we can add \(7x\) to both sides to simplify further:
\[
6 - 7x + 7x = 1 - 7x + 7x.
\]
This simplifies to:
\[
6 = 1.
\]
**Step 4: Analyze the result**
At this point, we see that the equation \(6 = 1\) is not true. This indicates that there are no values of \(x\) that can satisfy the original equation.
**Conclusion:**
The original equation has no solutions. In summary, we applied the properties of equality to manipulate the equation step by step and found that it leads to a contradiction. Thus, the number of solutions to the equation is:
\[
\boxed{0}.
\]