To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), we'll justify each step using properties of equality and determine the number of solutions. Let's go through it step by step.
Step 1: Distribute \( \frac{3}{2} \) on the left side.
Using the distributive property: \[ \frac{3}{2}(4) - \frac{3}{2}(8x) + 5x = 1 - 7x \] Calculating \( \frac{3}{2}(4) \): \[ 6 \] Calculating \( \frac{3}{2}(8x) \): \[ 12x \] Thus, we have: \[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side.
Combine \( -12x + 5x \): \[ 6 - 7x = 1 - 7x \]
Step 3: Isolate the variable \( x \).
Add \( 7x \) to both sides to eliminate \( -7x \) on the left side: \[ 6 = 1 \]
Step 4: Analyze the result.
The equation \( 6 = 1 \) is a false statement. This means there are no values for \( x \) that will satisfy the equation.
Conclusion:
The original equation has no solution. In conclusion, when you end up with a false statement while trying to isolate the variable, it indicates that the equation is inconsistent. Thus, the number of solutions to the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) is zero.
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