Question

then transition into just writing inequalities (skip part b,c) → go over inequality symbols first!!! https://www.jmap.org/worksheets/a.ced.a.3.modelingsystemsoflinearinequalities.pdf writing inequations: a high school drama club is putting on their annual theater production. there is a maximum of 800 tickets for the show. the costs of the tickets are $6 before the day of the show and $9 on the day of the show. to meet the expenses of the show, the club must sell at least $5,000 worth of tickets. a. write a system of inequalities that represent this situation. b. the club sells 440 tickets before the day of the show. is it possible to sell enough additional tickets on the day of the show to at least meet the expenses of the show? justify your answer. a drama club is selling tickets to the spring musical. the auditorium holds 200 people. tickets cost $12 at the door and $8.50 if purchased in advance. the drama club has a goal of selling at least $1000 worth of tickets to saturdays show. a. write a system of inequalities that can be used to model this scenario. b. if 50 tickets are sold in advance, what is the minimum number of tickets that must be sold at the door so that the club meets its goal? justify your answer. the drama club is running a lemonade stand to raise money for its new production. a local grocery store donated cans of lemonade and bottles of water. cans of lemonade sell for $2 each and bottles of water sell for $1.50 each. the club needs to raise at least $500 to cover the cost of renting costumes. the students can accept a maximum of 360 cans and bottles. a. write a system of inequalities that can be used to represent this situation. b. the club sells 144 cans of lemonade. what is the least number of bottles of water that must be sold to cover the cost of renting costumes? justify your answer. an on-line electronics store must sell at least $2500 worth of printers and computers per day. each printer costs $50 and each computer costs $500. the store can ship a maximum of 15 items per day. a. write a system of inequalities that can be used to represent this situation. b. on the set of axes below, graph a system of inequalities that models these constraints. c. determine a combination of printers and computers that would allow the electronics store to meet all of the constraints. explain how you obtained your answer. sarah earns $6 per hour babysitting and $12 per hour tutoring. her goal is to earn at least $120 per week. sarah is allowed to work a maximum of 14 hours per week doing both jobs. a. if x represents the number of hours sarah babysits and y represents the number of hours she tutors, write a system of inequalities that could model this situation. b. on the set of axes below, graph the system of inequalities that you wrote. c. state a combination of hours babysitting and tutoring that would satisfy this situation. justify your answer.

Explanation:

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First Scenario (High School Drama Club)

Step1: Define variables

Let $x$ = number of pre-show tickets, $y$ = number of day-of-show tickets.

Step2: Set ticket limit inequality

Total tickets ≤ 800: $x + y \leq 800$

Step3: Set revenue minimum inequality

Total revenue ≥ $5000$: $6x + 9y \geq 5000$

Step4: Set non-negative ticket constraints

$x \geq 0$, $y \geq 0$

Step5: Test 440 pre-show tickets

Substitute $x=440$ into revenue inequality:
$6(440) + 9y \geq 5000$
$2640 + 9y \geq 5000$
$9y \geq 2360$
$y \geq \frac{2360}{9} \approx 262.22$
Max remaining tickets: $800-440=360$, which is ≥263.
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Second Scenario (Spring Musical)

Step1: Define variables

Let $a$ = advance tickets, $d$ = door tickets.

Step2: Set venue capacity inequality

$a + d \leq 200$

Step3: Set revenue minimum inequality

$8.50a + 12d \geq 1000$

Step4: Set non-negative ticket constraints

$a \geq 0$, $d \geq 0$

Step5: Test 50 advance tickets

Substitute $a=50$:
$8.50(50) + 12d \geq 1000$
$425 + 12d \geq 1000$
$12d \geq 575$
$d \geq \frac{575}{12} \approx 47.92$
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Third Scenario (Lemonade Stand)

Step1: Define variables

Let $c$ = lemonade cans, $w$ = water bottles.

Step2: Set item limit inequality

$c + w \leq 360$

Step3: Set revenue minimum inequality

$2c + 1.50w \geq 500$

Step4: Set non-negative item constraints

$c \geq 0$, $w \geq 0$

Step5: Test 144 lemonade cans

Substitute $c=144$:
$2(144) + 1.50w \geq 500$
$288 + 1.50w \geq 500$
$1.50w \geq 212$
$w \geq \frac{212}{1.50} \approx 141.33$
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Fourth Scenario (Electronics Store)

Step1: Define variables

Let $p$ = printers, $c$ = computers.

Step2: Set shipping limit inequality

$p + c \leq 15$

Step3: Set revenue minimum inequality

$50p + 500c \geq 2500$ (simplify to $p + 10c \geq 50$)

Step4: Set non-negative item constraints

$p \geq 0$, $c \geq 0$

Step5: Find valid combination

Test $c=5$, $p=0$:
$0 + 5 \leq 15$, $50(0) + 500(5)=2500 \geq 2500$
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Fifth Scenario (Sarah's Jobs)

Step1: Use given variables

$x$ = babysitting hours, $y$ = tutoring hours.

Step2: Set total hour limit inequality

$x + y \leq 14$

Step3: Set earnings minimum inequality

$6x + 12y \geq 120$ (simplify to $x + 2y \geq 20$)

Step4: Set non-negative hour constraints

$x \geq 0$, $y \geq 0$

Step5: Find valid hour combination

Test $x=8$, $y=6$:
$8+6=14 \leq14$, $6(8)+12(6)=48+72=120 \geq120$
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Answer:

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First Scenario

a. System of inequalities:

$$\begin{cases} x + y \leq 800 \\ 6x + 9y \geq 5000 \\ x \geq 0 \\ y \geq 0 \end{cases}$$

b. Yes, it is possible. The club needs to sell at least 263 day-of-show tickets, and 360 tickets are still available, which is more than enough.
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Second Scenario

a. System of inequalities:

$$\begin{cases} a + d \leq 200 \\ 8.50a + 12d \geq 1000 \\ a \geq 0 \\ d \geq 0 \end{cases}$$

b. The minimum number of door tickets is 48.
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Third Scenario

a. System of inequalities:

$$\begin{cases} c + w \leq 360 \\ 2c + 1.50w \geq 500 \\ c \geq 0 \\ w \geq 0 \end{cases}$$

b. The least number of water bottles is 142.
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Fourth Scenario

a. System of inequalities:

$$\begin{cases} p + c \leq 15 \\ p + 10c \geq 50 \\ p \geq 0 \\ c \geq 0 \end{cases}$$

c. A valid combination is 0 printers and 5 computers. This meets the shipping limit ($0+5=15 \leq15$) and the revenue requirement ($50(0)+500(5)=2500 \geq2500$).
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Fifth Scenario

a. System of inequalities:

$$\begin{cases} x + y \leq 14 \\ x + 2y \geq 20 \\ x \geq 0 \\ y \geq 0 \end{cases}$$

c. A valid combination is 8 hours babysitting and 6 hours tutoring. This uses the maximum allowed hours ($8+6=14$) and meets the earnings goal ($6(8)+12(6)=120$).