Question

1. karen wants to buy a photo album for $16.25 and as many clear plastic sleeves as she can while spending no more than $25. the clear plastic sleeves cost $0.85 each. write an inequality that represents the relationship between the cost of the photo album and plastic sleeves karen wants to buy and the amount of money she has available to spend. cost of photo album cost of 1 sleeve number of sleeves money available 16.25 + 0.85c ≤ 25
2. nick wants to fence in a rectangular section of his yard to make a dog - run with a perimeter of at least 48 ft. the total length of the two shorter sides will be 16 ft. write an inequality to represent the relationship between the length ℓ of each of the longer sides of the dog - run and its perimeter.
3. miranda earns $475 a week plus a bonus of $150 for each machine she sells. her goal is to earn at least $1,000 a week. write an inequality to represent the relationship between mirandas goal and the number of machines she sells a week.
4. preston wants to earn a score of at least 90 on the test he is taking. he knows he got the first question, which is worth 25 points, correctly. the remaining questions are worth 3 points each. write an inequality that represents the relationship between his desired test score and the number of 3 - point questions q that he answers correctly.
5. reasoning how would the inequality symbol in exercises 6 - 8 change if the words at least were changed to at most? how would the meaning of the inequality change if the words at least were changed to at most?

Explanation:

Step1: Analyze Karen's situation

Let $c$ be the number of clear - plastic sleeves. The cost of the photo album is $16.25$ and the cost of each sleeve is $0.85$. The total amount she can spend is no more than $25$. So the inequality is $16.25 + 0.85c\leq25$.

Step2: Analyze Nick's situation

Let $\ell$ be the length of each of the longer sides of the rectangle. The total length of the two shorter sides is $16$ ft and the perimeter $P$ of a rectangle is $P = 2\ell+16$. The perimeter should be at least $48$ ft. So the inequality is $2\ell + 16\geq48$.

Step3: Analyze Miranda's situation

Let $n$ be the number of sales. Miranda earns $475$ per week plus a $150$ - dollar bonus for each sale. Her goal is to earn at least $1000$ per week. So the inequality is $475+150n\geq1000$.

Step4: Analyze Preston's situation

Let $q$ be the number of 3 - point questions answered correctly. Preston gets $25$ points for the first question and $3$ points for each of the remaining questions. He wants to earn at least $90$ points. So the inequality is $25 + 3q\geq90$.

Step5: Analyze the change of inequality symbols

If "at least" ($\geq$) is changed to "at most" ($\leq$), the direction of the inequality sign is reversed. For example, if in the inequality $25 + 3q\geq90$ (Preston's situation), if the goal is changed to "at most 90 points", the inequality becomes $25 + 3q\leq90$.

Answer:

1. $16.25 + 0.85c\leq25$
2. $2\ell + 16\geq48$
3. $475+150n\geq1000$
4. $25 + 3q\geq90$
5. If "at least" is changed to "at most", the inequality sign changes from $\geq$ to $\leq$.