Question

When packing a sultcase for a trip, the optimal welght of the suttcase Is 40 pounds. You aim for this weight with every trip you take. But your

sultcase's actual welght tends to vary by at most a certain number of pounds.

Part A. Write an absolute value inequality that models this relationship.

Part B. If the suitcase weight can vary by at most 7.5 pounds,what inequality can be used to find the range of acceptable weights for your suitcase?

Part C. Use the inequality from part B to determine the range of acceptable weights for the suitcase.

Part E. How would the inequality in part B change if you wanted to know the rate of weights that are unacceptable for your suitcase? Explain.

Part F. What is the solution for the inequality you created in part E?

Part H. What comparisons can you make between the two inequalities?

Part I. Will there ever be an instance where there is no solution to an absolute inequality? If yes,then what would the inequality look like?

Answer 1

To model the variation in a suitcase's weight, which is ideally 40 pounds, we use an absolute value inequality. For a variation of 7.5 pounds, the inequality is |W - 40| < 7.5, leading to an acceptable range of 32.5 to 47.5 pounds. The inequality for unacceptable weights is |W - 40| > 7.5, which highlights weights below 32.5 and above 47.5 pounds.

Step-by-step explanation:

The aim is to write an absolute value inequality that models the weight of a suitcase that ideally weighs 40 pounds but can have a variation in its weight.

Part A

The general form of an absolute value inequality for the suitcase weight W is |W - 40| ≤ d, where d is the maximum variation from the optimal weight.

Part B

If the weight can vary by at most 7.5 pounds, the inequality becomes |W - 40| ≤ 7.5.

Part C

To find the range of acceptable weights: 32.5 ≤ W ≤ 47.5.

Part E

To find the range of unacceptable weights, the inequality flips: |W - 40| > 7.5.

Part F

The unacceptable weight range: W < 32.5 or W > 47.5.

Part H

The comparison between the two inequalities demonstrates that one gives the range of acceptable weights while the other provides the range of unacceptable weights. They are complementary about the permissible variation from the optimal weight.

Part I

An absolute value inequality with no solution would be if the inequality imposes a condition that can never be true, such as |W - 40| < -d, which is impossible since absolute values cannot be negative.