To identify which property is not needed to justify any of the steps in Antonio's solution, let's analyze each step:
- 1.2x – 5.12 = 1.6 + 9(0.1x): This is the original equation.
- 1.2x – 5.12 = 1.6 + 0.9x: This step involves simplifying \(9(0.1x)\) to \(0.9x\), which does not require a property of equality since it's a simplification.
- 0.3x – 5.12 = 1.6: To obtain this, he likely moved \(0.9x\) from the right side to the left side by subtraction. This step requires the subtraction property of equality.
- 0.3x = 6.72: This was likely done by adding \(5.12\) to both sides, so it requires the addition property of equality.
- x = 22.4: This step involves division by \(0.3\), which requires the division property of equality.
After examining the steps, the step that does not require any of these properties is the second one, when he simplified \(9(0.1x)\) to \(0.9x\).
The only mentioned property that is not needed to justify any of the steps, therefore, is the associative property of addition. This property is related to how numbers are grouped in addition but is not necessary for simplifications or solving equations.
Thus, the answer is:
associative property of addition.