To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), let's start by simplifying and justifying the steps using properties of equality.
Step 1: Distribute \( \frac{3}{2} \)
Using the distributive property: \[ \frac{3}{2}(4) + \frac{3}{2}(-8x) + 5x = 1 - 7x \] This simplifies to: \[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side
Combine the \( x \) terms: \[ 6 - 12x + 5x = 6 - 7x \] So we have: \[ 6 - 7x = 1 - 7x \]
Step 3: Isolate the constant terms
To isolate the constant, we can subtract \( 1 \) from both sides: \[ 6 - 1 - 7x = -7x \] This simplifies to: \[ 5 - 7x = -7x \]
Step 4: Add \( 7x \) to both sides
Using the addition property of equality, we add \( 7x \) to both sides: \[ 5 - 7x + 7x = -7x + 7x \] This simplifies to: \[ 5 = 0 \]
Step 5: Interpretation
The statement \( 5 = 0 \) is a contradiction. This indicates that there are no values of \( x \) that would satisfy the original equation.
Conclusion
The original equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) has no solutions. This conclusion is based on the fact that we arrived at a contradictory statement, indicating that the necessary condition for any \( x \) is impossible.
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