Let's solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) step by step, using properties of equality and justifying each step.
Step 1: Distribute the \( \frac{3}{2} \) \[ \frac{3}{2}(4 - 8x) = \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x = 6 - 12x \]
So, the equation becomes: \[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side Combine \( -12x \) and \( 5x \): \[ 6 - 7x = 1 - 7x \]
Step 3: Add \( 7x \) to both sides Using the property of equality (adding the same quantity to both sides): \[ 6 = 1 \]
Step 4: Analyze the result The equation \( 6 = 1 \) is not true. This result indicates that there are no values of \( x \) that satisfy the original equation.
Conclusion: The equation has no solutions. This situation is derived from the fact that we reached a false statement after simplification, which indicates that the original equation is inconsistent. Thus, the number of solutions is: \[ \text{No solutions} \]
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