Inequalities

\(\begin{aligned}
&\text{The shaded region shown represents the solutions to }rx+ty\geq-77,\text{ where }r\text{ and }t\text{ are}\\
&\text{constants. What is the value of }r+t?
\end{aligned}\)

\(\begin{aligned}&\text{In the }xy\text{-plane, which of the following does }\text{NOT contain any points }(x,y)\text{ that are}\\&\text{solutions to }7x+4y>12?& \end{aligned}\)

\(\begin{aligned}&\text{The total cost, in dollars, to rent a surfboard consists of a \$25 service fee and a \$10}\\&\text{per hour rental fee. A person rents a surfboard for }t\text{ hours and intends to spend a}\\&\text{maximum of \$75 to rent the surfboard. Which inequality represents this situation?}
\end{aligned}\)

\(\begin{aligned}
&\text{For a snowstorm in a certain town, the minimum rate of snowfall recorded was 0.6}\\
&\text{inches per hour, and the maximum rate of snowfall recorded was 1.8 inches per hour. }\\
&\text{Which inequality is true for all values of }s,\text{ where }s\text{ represents a rate of snowfall, in}\\
&\text{inches per hour, recorded for this snowstorm?}
\end{aligned}\)

\(\begin{aligned}
&\text{A botanist estimates the number of oak trees in a forest to be at least 2600 and}\\
&\text{no more than 3200. Which inequality best represents the number oak trees, }x\\
&\text{in the forest?}
\end{aligned}\)

\(\begin{aligned}&\text{Cristina was elected to the city council. Each city council person is required to attend at least}\\&\text{21 of the 27 scheduled council meetings in a year to keep their position. Which inequality}\\&\text{represents the number of city council meetings, }x\text{, Cristina must attend in a year to keep her}\\&\text{position on the city council?}\end{aligned}\)

\(\begin{aligned}&\text{A plant nursery classifies a perennial flower species as tall if its typical height when fully}\\&\text{grown is more than 100 centimeters. Each of the following inequalities represents the}\\&\text{possible heights }h,\text{ in centimeters, for a specific perennial flower species when fully grown.}\\&\text{Which inequality represents the possible heights }h\text{, in centimeters, for a tall perennial}\\&\text{flower species?}\end{aligned}\)

\(\begin{aligned}&\text{When several resistors are connected in series in a circuit, the total resistance of the circuit}\\&\text{is the sum of the resistances of these resistors. A certain circuit consists of 8 resistors}\\&\text{connected in series, where the resistance of each resistor is positive. This circuit has a total}\\&\text{resistance of 130 ohms. The total resistance of 3 resistors in this circuit is 90 ohms. Which}\\&\text{inequality best represents all possible values of the resistance }x,\text{in ohms, of one of the other}\\&\text{5 resistors in this circuit?}\end{aligned}\)

\(\begin{aligned}&\text{A model estimates that a gray whale travels 72 to 77 miles each day during its migration.}\\&\text{Based on this model, which inequality represents the estimated total number of miles, }x,\text{a}\\&\text{gray whale could travel in 14 days of its migration?}\end{aligned}\)

\(\begin{aligned}&\text{A geologist estimates that the volume of a boulder is greater than 14.6 cubic feet but less}\\&\text{than 18.6 cubic feet. The geologist also estimates that the boulder weighs 160 pounds per}\\&\text{cubic foot of volume. Which inequality represents this situation, where }x\text{ represents the}\\&\text{estimated total weight, in pounds, of the boulder?}\end{aligned}\)

\(\begin{aligned}&\text{A shopper goes to a pet store to buy dog food and treats. The shopper has \$20. Each can of}\\&\text{food costs \$1.87, and each bag of treats costs \$2.37. Which inequality represents the}\\&\text{relationship between the number of cans of food, }c, \text{and the number of bags of treats, }t,\text{ the}\\&\text{shopper can buy?}\end{aligned}\)

\(\begin{aligned}&\text{A newspaper stand sells magazines for \$5 each and newspapers for \$2 each. The stand}\\&\text{needs to sell }m\text{ magazines and }n\text{ newspapers for a combined total value of at least \$150}\\&\text{each day to cover operational costs. Which inequality represents this situation?}\end{aligned}\)

\(\begin{aligned}&\text{To win a game show, a contestant needs to score at least 100 total points from two rounds.}\\&\text{Correct responses in the first round are worth 2 points each, and correct responses in the}\\&\text{second round are worth 5 points each. Which inequality models this situation, where }f\text{ is}\\&\text{the number of correct responses in the first round and }s\text{ is the number of correct}\\&\text{responses in the second round?}\end{aligned}\)

\(\begin{aligned}&\text{To cut a lawn, Antwan charges a fee of \$10.00 for his equipment and \$8.50 per hour spent}\\&\text{cutting a lawn. Taylor charges a fee of \$7.00 for her equipment and \$9.50 per hour spent}\\&\text{cutting a lawn. If }x\text{ represents the number of hours spent cutting a lawn, what are all the}\\&\text{values of }x\text{ for which Taylor’s total charge is greater than Antwan’s total charge?}\end{aligned}\)

\(\begin{aligned}&\text{Ty set a goal to walk at least 24 kilometers every day to prepare for a multiday hike.}\\&\text{On a certain day, Ty plans to walk at an average speed of 4 kilometer per hour. What}\\&\text{is the minimum number of hours Ty must walk on that day to fulfill the daily goal?}\end{aligned}\)

\(\begin{aligned}&\text{An event planner is planning a party. It costs the event planner a onetime fee of \$35 to}\\
&\text{rent the venue and \$10.25 per attendee. The event planner has a budget of \$200. What}\\
&\text{is the greatest number of attendees possible without exceeding the budget?}\end{aligned}\)

\(\begin{aligned}
&\text{Maria plans to rent a boat. The boat rental costs \$60 per hour, and she will also have to}\\
&\text{pay for a water safety course that costs \$10. Maria wants to spend no more than \$280 for}\\
&\text{the rental and the course. If the boat rental is available only for a whole number of hours,}\\
&\text{what is the maximum number of hours for which Maria can rent the boat?}\end{aligned}\)

\(\begin{aligned}
&\text{The posted weight limit for a covered wooden bridge in Pennsylvania is 6000 pounds.}\\
&\text{A delivery truck that is carrying }x\text{ identical boxes each weighing 14 pounds will pass}\\
&\text{over the bridge. If the combined weight of the empty delivery truck and its driver is}\\
&\text{4500 pounds, what is the maximum possible value for }x\text{ that will keep the combined }\\
&\text{weight of the truck, driver, and boxes below the bridge’s posted weight limit?}\\
\end{aligned}\)

\(\begin{aligned}
&\text{A moving truck can tow a trailer if the combined weight of the trailer and the boxes}\\
&\text{it contains is no more than 4,600 pounds. What is the maximum number of boxes this}\\
&\text{truck can tow in a trailer with weight of 500 pounds if each box weighs 120 pounds?}\end{aligned}\)

\(\begin{aligned}
&\text{Tickets for a school talent show cost \$2 for students and \$3 for adults. If Chris}\\
&\text{spends at least \$11 but no more than \$14 on }x\text{ student tickets and 1 adult ticket,}\\
&\text{what is one possible value of }x?
\end{aligned}\)

\(\begin{aligned}
&\text{A certain blue whale calf weighed 5,600 pounds when it was born, and its weight increased}\\
&\text{at an average rate of 250 pounds per day for the first }n\text{ days after birth, where }n\text{ is an}\\
&\text{integer. If the calf’s weight was greater than 8,850 pounds but less than 9,850 pounds }n\text{ days}\\
&\text{after birth, what is one possible value of }n?\end{aligned}\)

\(\begin{aligned}
&\text{Lynne has \$8.00 to spend on apples and oranges. Apples cost \$0.65 each, and oranges}\\
&\text{cost \$0.75 each. If there is no tax on this purchase and she buys 5 apples, what is the}\\
&\text{maximum number of whole oranges she can buy}?\end{aligned}\)

\(\begin{aligned}
&\text{An employee earns \$30 per hour for the first 40 hours worked in a week. For each hour}\\
&\text{worked over 40 hours, the employee earns \$45 per hour. What is the least number of whole}\\
&\text{hours that the employee must work in a week to earn at least \$1,610}?
\end{aligned}\)

\(\begin{aligned}
&\text{If }x \geq-6 \text{ represents all solutions to the inequality }ax-27\leq15,\text{ where }a\text{ is a constant,}\\
&\text{what is the greatest possible value of }a?
\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y>14\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4x+y<18\\
&\text{The point }(x,53)\text{ is a solution to the system of inequalities in the }xy\text{-plane. }\\
&\text{Which of the following could be the value of }x?
\end{aligned}\)

\(\begin{aligned}&
\text{The point (8, 2) in the }xy\text{-plane is a solution to which of the following }\\&\text{systems of inequalities?}
\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y\leq x+7\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y\geq-2x-1\\
&\text{Which point }(x,y)\text{ is a solution to the given system of inequalities in the }xy\text{-plane?}\\
\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y\leq 3x+1\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x-y>1\\
&\text{Which point }(x,y)\text{ is a solution to the given system of inequalities in the }xy\text{-plane?}\\
\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y<x\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x<22\\
&\text{For which of the following tables are all the values of }x\text{ and their corresponding values of } y\\
&\text{solutions to the given system of inequalities?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y<-5x-19\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y>-3x-12\\
&\text{For which of the following tables are all the values of }x\text{ and their corresponding values of}\\
&y\text{ solutions to the given system of inequalities?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y<24-6x\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{y}{6}>9\\
&\text{Which inequality represents the }x\text{ values for all solutions }(x,y)\text{ that satisfy the given system}\\
&\text{of inequalities in the }xy\text{-plane}?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y>2x-1\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2x>5\\
&\text{Which inequality represents the }y\text{ values for all solutions }(x,y)\text{ that satisfy the given system}\\
&\text{of inequalities in the }xy\text{-plane}?\end{aligned}\)

\(\begin{aligned}
&\text{If the system of inequalities }y\geq2x+1\text{ and }y>\frac{1}{2}x-1\text{ is graphed in the }xy\text{-plane }\\
&\text{above, which quadrant contains no solutions to the system?}
\end{aligned}\)

\(\begin{aligned}
&\text{A worker uses a forklift to move boxes that weigh either 40 pounds or 65 pounds each.}\\
&\text{Let }x\text{ be the number of 40-pound boxes and }y\text{ be the number of 65-pound boxes. The}\\
&\text{forklift can carry up to either 45 boxes or a weight of 2,400 pounds. Which of the}\\
&\text{following systems of inequalities represents this relationship?}\end{aligned}\)

\(\begin{aligned}
&\text{Jackie has two summer jobs. She works as a tutor, which pays \$12 per hour, and she works }\\
&\text{as a lifeguard, which pays \$9.50 per hour. She can work no more than 20 hours per week, but }\\
&\text{she wants to earn at least \$220 per week. Which of the following systems of inequalities }\\
&\text{represents this situation in terms of }x\text{ and }y,\text{ where }x\text{ is the number of hours she tutors and }y\\
&\text{is the number of hours she works as a lifeguard?}\end{aligned}\)

\(\begin{aligned}&\text{A landscaper will plant two types of shrubs, hydrangeas and winter hazels, in a park. There}\\&\text{will be no more than 244 total shrubs planted, and the number of hydrangeas planted will}\\&\text{be at most three times the number of winter hazels planted. Which of the following}\\&\text{systems of inequalities best represents this situation, where }h\text{ is the number of hydrangeas}\\&\text{that will be planted and }w\text{ is the number of winter hazels that will be planted?}\end{aligned}\)

\(\begin{aligned}
&\text{A laundry service is buying detergent and fabric softener from its supplier. The supplier will}\\
&\text{deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds, }\\
&\text{and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice }\\
&\text{as many containers of detergent as containers of fabric softener. Let }d\text{ represent the number of }\\
&\text{containers of detergent, and let }s\text{ represent the number of containers of fabric}\text{ softener, where }\\
&d\text{ and }s\text{ are nonnegative integers. Which of the following systems of inequalities best represents}\\&\text{this situation?}\end{aligned}\)

\(\begin{aligned}&\text{A biologist is designing a study to observe the behavior of male and female amethyst-}\\&\text{throated hummingbirds. According to the study’s design, at least 91 hummingbirds will be}\\&\text{observed, and the positive difference between the number of males, }x,\text{and the number of}\\&\text{females, }y\text{, will not exceed 8. Which of the following systems of inequalities represents this}\\&\text{situation?}\end{aligned}\)

\(\begin{aligned}
&\text{A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each }\\
&\text{delivery trip, the helicopter must carry at least 10 packages, and the total weight of the}\\
&\text{packages can be at most 1,100 pounds. What is the maximum number of 120-pound}\\
&\text{packages that the helicopter can carry per trip?}\end{aligned}\)

\(\begin{aligned}
&\text{A small business owner budgets \$2,200 to purchase candles.The owner must purchase a }\\
&\text{minimum of 200 candles to maintain the discounted pricing. If the owner pays \$4.90 per }\\
&\text{candle to purchase small candles and \$11.60 per candle to purchase large candles, what }\\
&\text{is the maximum number of large candles the owner can purchase to stay within the budget }\\
&\text{and maintain the discounted pricing?}\end{aligned}\)