Question

directions
read the instructions for this self - checked activity. type in your response to each question, and check your answers. at the end of the activity, write a brief evaluation of your work.
activity
in this activity, youll write and solve an absolute value inequality in context. read the scenario, and answer the questions that follow.
when packing a suitcase for a trip, the optimal weight of the suitcase is 40 pounds. you aim for this weight with every trip you take, but your suitcases actual weight tends to vary.
part a
the rules state that the weight of the suitcase can vary by at most 7.5 pounds. write an inequality you could use to find the range of acceptable weights for your suitcase, where x is the weight of your suitcase.
drag each term to the correct location in the image.
image of a drag - and - drop interface with terms: x + 40, x - 40, 7.5, 40, x - 7.5, ≥, ≤
submit
part b
which inequality describes the acceptable weights for the suitcase?
○ x ≤ 32.5 or x ≤ 47.5
○ 32.5 ≥ x ≥ 47.5
○ x ≤ 32.5 or x ≥ 47.5
○ 32.5 ≤ x ≤ 47.5

Explanation:

Step1: Define absolute value inequality

The absolute difference between actual weight $x$ and optimal weight 40 is at most 7.5, so the structure is $|x-40| \leq 7.5$.

Step2: Solve the absolute value inequality

Split into two inequalities:
$x-40 \leq 7.5$ and $x-40 \geq -7.5$

Step3: Calculate bounds

For $x-40 \leq 7.5$: $x \leq 40+7.5 = 47.5$
For $x-40 \geq -7.5$: $x \geq 40-7.5 = 32.5$
Combine to get $32.5 \leq x \leq 47.5$.

Answer:

Part A:

$|x-40| \leq 7.5$

Part B:

D. $32.5 \leq x \leq 47.5$