Answer :
Let's analyze Michael's steps carefully to understand the properties used in each step. Our focus is between Step 3 and Step 4.
Michael's work:
Step 1: [tex]\(-6(x+3) + 10 < -2\)[/tex]
This step applies the expression as it is given.
Step 2: [tex]\(-6x - 18 + 10 < -2\)[/tex]
In this step, Michael applied the distribution property to distribute [tex]\(-6\)[/tex] across the terms inside the parentheses: [tex]\(-6 \times x\)[/tex] and [tex]\(-6 \times 3\)[/tex].
Step 3: [tex]\(-6x - 8 < -2\)[/tex]
Here, Michael combined like terms by simplifying [tex]\(-18 + 10\)[/tex] to get [tex]\(-8\)[/tex].
Step 4: [tex]\(-6x < 6\)[/tex]
In this step, Michael added [tex]\(8\)[/tex] to both sides of the inequality to isolate the term containing [tex]\(x\)[/tex].
To isolate [tex]\(x\)[/tex] on one side of the inequality, Michael performed the following operation:
[tex]\[ -6x - 8 + 8 < -2 + 8 \][/tex]
[tex]\[ -6x < 6 \][/tex]
This operation is justified by the addition property of inequality, which states that adding the same amount to both sides of an inequality does not change the direction of the inequality.
Step 5: [tex]\(x > -1\)[/tex]
Finally, Michael divided both sides by [tex]\(-6\)[/tex] and reversed the inequality sign because of the division by a negative number.
To summarize, the correct property that justifies the work shown between Step 3 and Step 4 is the addition property of inequality.
Thus, the correct answer is:
[tex]\[ \boxed{\text{D. addition property of inequality}} \][/tex]
Michael's work:
Step 1: [tex]\(-6(x+3) + 10 < -2\)[/tex]
This step applies the expression as it is given.
Step 2: [tex]\(-6x - 18 + 10 < -2\)[/tex]
In this step, Michael applied the distribution property to distribute [tex]\(-6\)[/tex] across the terms inside the parentheses: [tex]\(-6 \times x\)[/tex] and [tex]\(-6 \times 3\)[/tex].
Step 3: [tex]\(-6x - 8 < -2\)[/tex]
Here, Michael combined like terms by simplifying [tex]\(-18 + 10\)[/tex] to get [tex]\(-8\)[/tex].
Step 4: [tex]\(-6x < 6\)[/tex]
In this step, Michael added [tex]\(8\)[/tex] to both sides of the inequality to isolate the term containing [tex]\(x\)[/tex].
To isolate [tex]\(x\)[/tex] on one side of the inequality, Michael performed the following operation:
[tex]\[ -6x - 8 + 8 < -2 + 8 \][/tex]
[tex]\[ -6x < 6 \][/tex]
This operation is justified by the addition property of inequality, which states that adding the same amount to both sides of an inequality does not change the direction of the inequality.
Step 5: [tex]\(x > -1\)[/tex]
Finally, Michael divided both sides by [tex]\(-6\)[/tex] and reversed the inequality sign because of the division by a negative number.
To summarize, the correct property that justifies the work shown between Step 3 and Step 4 is the addition property of inequality.
Thus, the correct answer is:
[tex]\[ \boxed{\text{D. addition property of inequality}} \][/tex]
