Inequality Modeling

thunc14

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A company's goal is that every month, they will sell 10% more than what was sold the previous year. The CEO considers it a success if they can be within 10% of that goal. The previous year's sales for April totaled $20,000

Which inequality describes the situation.
a. |x-2,000| ≤ 22,000
b. |x-22,000|≤ 2,200
c. |x-2,200| ≤ 22,000
d. |x-22,000| ≤ 2,000
e. None of the above

My answer would be e, because you could plug in numbers, such as 0 or numbers that are outside of the 10% increase and still make the inequalities true. Am I overlooking something or not understanding the questions correctly?
 

Dr.Peterson

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Please demonstrate what you are saying. What numbers that don't meet the goal satisfy all the inequalities offered?

Certainly x=0 makes some of them false.

Then also show us how you would write an inequality for the problem.
 

thunc14

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I apologize, I misspoke. Clearly 0 makes some of the inequalities false. Since the problem states within 10%, a success would be considered any sales between 20,000 and 22,000, a 2,000 margin.
I would say A is wrong because you could put in 0 and get a true inequality
I would say B is wrong because you could put in 23,000 and get a true inequality, but that's outside the 10% threshold
I would say C is wrong for the same reasons as A
I would say D is wrong for the same reasons as B

To be honest I'm not sure how I would write this inequality using absolute values. I having a hard time understanding why they are necessary in this case because we are talking about values between 20,000 and 22,000, a positive 10% increase. Am I missing something here?
 

Dr.Peterson

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I think you missed the fact that 20,000 is the previous year's sales, so the goal this year is 1.10*20,000 = 22,000. Anything within 10% of that (in either direction) is acceptable.
 

thunc14

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Ah I see. So the inequality that models this would be D, since any value between 20,000 and 24,000 would satisfy this 10% (2000) on either side. Correct?
 

Dr.Peterson

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Check that. What is 10% of the goal, $22,000?
 

thunc14

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Check that. What is 10% of the goal, $22,000?
Ah, so the 10% boundary is in regards to the 22k, my mistake. So it would have to be B, less than or equal to 10% of 22k, 2,200.
 

Dr.Peterson

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Correct. The options each reflect a common error one could make, so you have to think very carefully before deciding that a match means you're right!
 

Jomo

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This problem is strange as the CEO will not consider a success if they increase by more than say 25%
 
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