To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) and justify the steps using properties of equality, let's follow through with the solution.
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Distribute \( \frac{3}{2} \) on the left side: \[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \] \[ 6 - 12x + 5x = 1 - 7x \] (Distributive property)
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Combine like terms on the left side: \[ 6 - 7x = 1 - 7x \] (Combining like terms)
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Add \( 7x \) to both sides: \[ 6 = 1 \] (Addition property of equality)
At this point, \( 6 = 1 \) is a contradiction and indicates that there is no solution to the equation.
- Conclusion: Since we arrived at a contradiction, this means that there are no values of \( x \) that satisfy the equation.
Thus, the number of solutions to the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) is 0 solutions.