To determine which property justifies the work between step 3 and step 4, let's analyze the steps in detail:
1. Step 1: [tex]\(-6(x+3) + 10 < -2\)[/tex]
2. Step 2: Distribute [tex]\(-6\)[/tex] through the parenthesis: [tex]\(-6x - 18 + 10 < -2\)[/tex].
3. Step 3: Combine like terms: [tex]\(-6x - 8 < -2\)[/tex].
4. Step 4: To isolate [tex]\(x\)[/tex], we add [tex]\(8\)[/tex] to both sides of the inequality:
[tex]\[
-6x - 8 + 8 < -2 + 8 \implies -6x < 6
\][/tex]
5. Step 5: Divide both sides by [tex]\(-6\)[/tex] and reverse the inequality sign (as required when dividing by a negative number):
[tex]\[
x > -1
\][/tex]
The crucial question is about the justification of the step between Step 3 and Step 4. From Step 3 to Step 4, [tex]\(8\)[/tex] is added to both sides of the inequality. The property that allows for adding the same value to both sides of an inequality without changing the direction of the inequality is:
D. Addition Property of Inequality
Thus, the addition property of inequality justifies the work shown between step 3 and step 4.
