To analyze the steps Antonio took to solve the equation, we can break down his reasoning:
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Starting Equation: \[ 1.2x - 5.12 = 1.6 + 9(0.1x) \] (Here, the distributive property is implicit when calculating \(9(0.1x)\))
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Simplification of the Right Side: \[ 1.2x - 5.12 = 1.6 + 0.9x \] (This uses the distributive property: \(9 \times 0.1 = 0.9\))
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Isolating the x Terms: \[ 1.2x - 0.9x - 5.12 = 1.6 + 0 - 0.9x \] (Here we subtract \(0.9x\) from both sides, using the property of equality)
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Rearranging: \[ 0.3x - 5.12 = 1.6 \]
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Adding 5.12 to Both Sides: \[ 0.3x = 1.6 + 5.12 \] (Again, this uses the property of equality)
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Combining the Right Side: \[ 0.3x = 6.72 \]
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Dividing by 0.3: \[ x = \frac{6.72}{0.3} \] (This uses the property of equality)
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Final Calculation: \[ x = 22.4 \]
Now, regarding the properties needed:
- The Distributive Property was used in the calculation of \(9(0.1x) = 0.9x\).
- The Subtraction Property of Equality (subtracting the same amount from both sides).
- The Addition Property of Equality (adding the same amount to both sides).
The Inversion Property (or multiplicative inverse) is not specifically utilized to justify any of the steps in the process since no steps required multiplying by reciprocal values in any part of solving the equation. Therefore, the property that is not needed to justify any of the steps is the Inversion Property (Multiplicative Inverse).