Let's analyze the steps taken by Michael to solve the inequality step by step and focus specifically on the transition from Step 3 to Step 4:
1. Step 1: [tex]\(-8(x + 3) + 10 < -2\)[/tex]
2. Step 2: [tex]\(-6x - 18 + 10 < -2\)[/tex]
- Here, the distribution property is used to expand [tex]\(-8(x + 3)\)[/tex] into [tex]\(-8 \cdot x + (-8) \cdot 3\)[/tex].
3. Step 3: [tex]\(-6x - 8 < -2\)[/tex]
- The terms [tex]\(-18\)[/tex] and [tex]\(10\)[/tex] are combined to get [tex]\(-8\)[/tex].
4. Step 4: [tex]\(-6x < 6\)[/tex]
- This step involves eliminating [tex]\(-8\)[/tex] from both sides of the inequality by adding [tex]\(8\)[/tex] to both sides. This results in [tex]\(-6x - 8 + 8 < -2 + 8\)[/tex], simplifying to [tex]\(-6x < 6\)[/tex].
5. Step 5: [tex]\(x > -1\)[/tex]
- This step employs the division property of inequality. However, our focus is on the step between 3 and 4.
To determine the property used between step 3 and step 4:
- In step 3: [tex]\(-6x - 8 < -2\)[/tex]
- In step 4: [tex]\(-6x < 6\)[/tex]
The change from step 3 to step 4 is obtained by adding [tex]\(8\)[/tex] to both sides of the inequality [tex]\(-6x - 8 < -2\)[/tex]. This addition does not affect the direction of the inequality since it is consistent on both sides.
Therefore, the property that justifies the transition from step 3 to step 4 is the addition property of inequality. This property states that adding the same number to both sides of an inequality does not change the inequality's direction.
Thus, the correct answer is:
D. addition property of inequality
