1. The length of a rectangle is four times its width. The perimeter of the rectangle is at most 130 cm.

Which inequality models the relationship between the width and the perimeter of the rectangle?

A)2w+2⋅(4w)>130

B)2w+2⋅(4w)<130

C)2w+2⋅(4w)≤130

D)2w+2⋅(4w)≥130

2. Which inequality models this problem?

Eduardo started a business selling sporting goods. He spent $7500 to obtain his merchandise, and it costs him $300 per week for general expenses. He earns $850 per week in sales.
A)850w≥7500+300w
B)300w>7500+850w
C)850w>7500+300w
D)850w<7500+300w

3. Jenny is eight years older than twice her cousin Sue’s age. The sum of their ages is less than 32.
What is the greatest age that Sue could be?
A)7
B)8
C)9
D)10

4. The sum of two consecutive integers is at most 223.
What is the larger of the two integers?

5. Vanessa has scored 45, 32, and 37 on her three math quizzes. She will take one more quiz and she wants a quiz average of at least 40.
What is the minimum score Vanessa needs to earn on her fourth quiz?

2. There is a statement of expenses and earnings in terms of weeks, w. There is no problem statement or question that is posed. C) will tell the number of weeks until profitability (earnings greater than expenses). A) will tell the number of weeks until Eduardo at least breaks even (earnings at least cover expenses).

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Comment on this question

Math is supposed to teach critical thinking and quantitative reasoning. The math curriculum these days often seems to ask questions about "this situation" without defining what "this situation" is. Such problems should be rejected by any critical thinker as being unanswerable.

Here, the description of Eduardo's business contains no inequality statement, so there is nothing that can be modeled by an inequality.

I like to refer questions like these to the teacher. Have your teacher show you the chain of reasoning—supported by statements or questions in the given problem—that gets you to the appropriate answer choice. (It can't be done, and your teacher should tell you as much.)

3. If s represents Sue's age, the problem asks for the largest possible value of s when ...

... s + (2s+8) < 32

... 3s < 24 . . . . . subtract 8; next, divide by 3

... s < 8

The largest integer less than 8 is 7, so the greatest age Sue could be is 7.

4. If n represents the largest integer then the constraint is ...

... (n-1) +n ≤ 223

... 2n ≤ 224 . . . . . . add 1; next, divide by 2

... n ≤ 112

5. I like to work these by considering the differences from the desired average. Vanessa's 3 scores differ from her desired average by 5, -8, and -3 points, a total of -6 points. To achieve her desired average, she needs to score at least 6 points higher than her desired average on her fourth quiz to bring that total to zero.

... 40 + 6 = 46 . . . . the minimum score Vanessa needs

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If you want an inequality, you can write one for the average of her 4 scores, where q represents her final quiz score.

... (45 +32 +37 +q)/4 ≥ 40

... 114 +q ≥ 160 . . . . multiply by 4

... q ≥ 46 . . . . . . . . subtract 114

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Equation for verbal solution

Effectively, we computed ...

... ((5 +40) +(-8 +40) +(-3 +40) +(∆q +40))/4 ≥ 40 . . . . . . where q = ∆q + 40

... (5 -8 -3 +∆q) ≥ 0 . . . . . multiply by 4, subtract 160 from both sides

... ∆q ≥ 6 . . . . . . . . . . . . . so, q ≥ 46