Select the correct answer.
Michael solved this inequality as shown:
Step 1: [tex]$-6(x+3)+10<-2$[/tex]
Step 2: [tex]$-6 x-18+10<-2$[/tex]
Step 3: [tex]$-6 x-8<-2$[/tex]
Step 4: [tex]$-6 x<6$[/tex]
Step 5: [tex]$x>-1$[/tex]
What property justifies the work shown between step 3 and step 4?
A. transitive property
B. division property of inequality
C. distribution property
D. addition property of inequality
Answer (2)
The problem requires identifying the property used to transition from − 6 x − 8 < − 2 to − 6 x < 6 .
Adding 8 to both sides of the inequality isolates the term with x .
The addition property of inequality justifies this step.
The correct answer is D. addition property of inequality.
Explanation
Analyze the Problem We are given an inequality that Michael solved step by step. Our task is to identify the property that justifies the transition from Step 3: − 6 x − 8 < − 2 to Step 4: − 6 x < 6 . The possible properties are transitive, division property of inequality, distribution property, and addition property of inequality.
Isolate the x term To go from Step 3 to Step 4, we need to isolate the term with x on one side of the inequality. This can be achieved by adding 8 to both sides of the inequality.
Apply the Addition Adding 8 to both sides of the inequality in Step 3 gives us: − 6 x − 8 + 8 < − 2 + 8 Simplifying this, we get: − 6 x < 6
Identify the Property The property that allows us to add the same number to both sides of an inequality without changing the validity of the inequality is the addition property of inequality.
State the Answer Therefore, the correct answer is D. addition property of inequality.
Examples
The addition property of inequality is useful in many real-world scenarios. For example, imagine you are saving money for a new bicycle that costs $200. You currently have 50 s a v e d . U s in g t h e a dd i t i o n p ro p er t yo f in e q u a l i t y , yo u c an d e t er min e h o w m u c hm ore m o n eyyo u n ee d t os a v e . I f x$ represents the additional amount you need to save, the inequality is 50 + x ≥ 200 . By adding − 50 to both sides, you find that x ≥ 150 . This means you need to save at least $150 more to afford the bicycle. This principle applies to budgeting, comparing costs, and many other practical situations.
To transition from Step 3 to Step 4, Michael used the addition property of inequality by adding 8 to both sides of the inequality. This step is important for isolating the term containing x . Thus, the correct answer is D. addition property of inequality.
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