PSAT: Algebra - Linear Inequalities in one or two variables

$y < 6x + 2$

For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given inequality?

A.

B.

C.

D.

The formula above is Ohm’s law for an electric circuit with current $I$, in amperes, potential difference $V$, in volts, and resistance $R$, in ohms.

A circuit has a resistance of 500 ohms, and its potential difference will be generated by $n$ six-volt batteries that produce a total potential difference of $6n$ volts.

If the circuit is to have a current of no more than 0.25 ampere, what is the greatest number, $n$, of six-volt batteries that can be used?

An event planner is planning a party. It costs the event planner a onetime fee of $\$35$ to rent the venue and $\$10.25$ per attendee. The event planner has a budget of $\$300$. What is the greatest number of attendees possible without exceeding the budget?

$y > 14$

$4x + y < 18$

The point $(x, 53)$ is a solution to the system of inequalities in the $xy$-plane. Which of the following could be the value of $x$?

A number $x$ is at most 2 less than 3 times the value of $y$. If the value of $y$ is $-4$, what is the greatest possible value of $x$?

A local transit company sells a monthly pass for $95 that allows an unlimited number of trips of any length. Tickets for individual trips cost $1.50, $2.50, or $3.50, depending on the length of the trip. What is the minimum number of trips per month for which a monthly pass could cost less than purchasing individual tickets for trips?

$y < x$

$x < 22$

For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given system of inequalities?

A.

B.

C.

D.

A number $x$ is at most 17 less than 5 times the value of $y$. If the value of $y$ is 3, what is the greatest possible value of $x$?

$y \le x$

$y \le -x$

Which of the following ordered pairs $(x, y)$ is a solution to the system of inequalities above?

A. $(1, 0)$

B. $(-1, 0)$

C. $(0, 1)$

D. $(0, -1)$

Adam’s school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, $w$, that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of $w$ for which it would be faster for Adam to walk to school?

A. $w - 5 < 20$

B. $w - 5 > 20$

C. $w + 5 < 20$

D. $w + 5 > 20$

A city employee will plant two types of bushes, azaleas and boxwoods, in a park. There will be no more than $164$ total bushes planted, and the number of azaleas planted will be at most three times the number of boxwoods planted. Which of the following systems of inequalities best represents this situation, where $a$ is the number of azaleas that will be planted, and $b$ is the number of boxwoods that will be planted?

A. $a + b \ge 164$
$3a \ge b$

B. $a + b \ge 164$
$a \le 3b$

C. $a + b \le 164$
$3a \ge b$

D. $a + b \le 164$
$a \le 3b$

The length of a rectangle is 50 inches and the width is $x$ inches. The perimeter is at most 210 inches. Which inequality represents this situation?

A. $2x + 100 \le 210$

B. $2x + 100 \ge 210$

C. $2x + 50 \le 210$

D. $2x + 50 \ge 210$

$y \le x + 7$

$y \ge -2x - 1$

Which point $(x, y)$ is a solution to the given system of inequalities in the $xy$-plane?

A. $(-14, 0)$

B. $(0, -14)$

C. $(0, 14)$

D. $(14, 0)$

The minimum value of $x$ is 12 less than 6 times another number $n$. Which inequality shows the possible values of $x$?

A. $x \le 6n - 12$

B. $x \ge 6n - 12$

C. $x \le 12 - 6n$

D. $x \ge 12 - 6n$

A moving truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than 4,600 pounds. What is the maximum number of boxes this truck can tow in a trailer with a weight of 500 pounds if each box weighs 120 pounds?

A. 34

B. 35

C. 38

D. 39

A business owner plans to purchase the same model of chair for each of the 81 employees. The total budget to spend on these chairs is $\$14{,}000$, which includes a $7\%$ sales tax. Which of the following is closest to the maximum possible price per chair, before sales tax, the business owner could pay based on this budget?

A. $\$148.15$

B. $\$161.53$

C. $\$172.84$

D. $\$184.94$

$y > 2x - 1$

$2x > 5$

Which of the following consists of the y-coordinates of all the points that satisfy the system of inequalities above?

A. $y > 6$

B. $y > 4$

C. $y > \dfrac{5}{2}$

D. $y > \dfrac{3}{2}$

$2\ell + 2w \le 27$

A rectangle has length $\ell$ and width $w$. The inequality gives the possible values of $\ell$ and $w$ for which the perimeter of this rectangle is less than or equal to 27. Which statement is the best interpretation of $(\ell, w) = (8, 3)$ in this context?

A. If the rectangle has length 3 and width 8, its perimeter is less than or equal to 27.

B. If the rectangle has length 8 and width 3, its perimeter is less than or equal to 27.

C. If the rectangle has length 3 and width 8, its perimeter is greater than or equal to 27.

D. If the rectangle has length 8 and width 3, its perimeter is greater than or equal to 27.

The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find $t$, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?

A. $25{,}000 > (4{,}334 - 2{,}712)t$

B. $25{,}000 < (4{,}334 - 2{,}712)t$

C. $25{,}000 - 4{,}334 > 2{,}712t$

D. $25{,}000 > \dfrac{4{,}332}{2{,}712} t$

A truck can haul a maximum weight of $5{,}630$ pounds. During one trip, the truck will be used to haul a $190$-pound piece of equipment as well as several crates. Some of these crates weigh $25$ pounds each and the others weigh $62$ pounds each. Which inequality represents the possible combinations of the number of $25$-pound crates, $x$, and the number of $62$-pound crates, $y$, the truck can haul during one trip if only the piece of equipment and the crates are being hauled?

A. $25x + 62y \le 5{,}440$

B. $25x + 62y \ge 5{,}440$

C. $62x + 25y \le 5{,}630$

D. $62x + 25y \ge 5{,}630$

An elementary school teacher is ordering $x$ workbooks and $y$ sets of flash cards for a math class. The teacher must order at least 20 items, but the total cost of the order must not be over $80. If the workbooks cost $3 each and the flash cards cost $4 per set, which of the following systems of inequalities models this situation?

A.

B.

C.

D.

$y \le 3x + 1$
$x - y > 1$

Which of the following ordered pairs (x, y) satisfies the system of inequalities above?

A. $(-2, -1)$

B. $(-1, 3)$

C. $(1, 5)$

D. $(2, -1)$

Ty set a goal to walk at least 24 kilometers every day to prepare for a multiday hike. On a certain day, Ty plans to walk at an average speed of 4 kilometers per hour. What is the minimum number of hours Ty must walk on that day to fulfill the daily goal?

A. 4

B. 6

C. 20

D. 24

A certain elephant weighs 200 pounds at birth and gains more than 2 but less than 3 pounds per day during its first year. Which of the following inequalities represents all possible weights $w$, in pounds, for the elephant 365 days after birth?

A. $400 < w < 600$

B. $565 < w < 930$

C. $730 < w < 1{,}095$

D. $930 < w < 1{,}295$

A bakery sells trays of cookies. Each tray contains at least 50 cookies but no more than 60. Which of the following could be the total number of cookies on 4 trays of cookies?

A. 165

B. 205

C. 245

D. 285

A geologist needs to collect at least 67 samples of lava from a volcano. If the geologist has already collected 63 samples from the volcano, what is the minimum number of additional samples the geologist needs to collect?

A. 130

B. 63

C. 4

D. 0

An event planner is planning a party. It costs the event planner a onetime fee of $35 to rent the venue and $10.25 per attendee. The event planner has a budget of $200. What is the greatest number of attendees possible without exceeding the budget?

$y < 5x + 6$

For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given inequality?

A.

B.

C.

D.

Which of the following ordered pairs $(x, y)$ satisfies the inequality $5x - 3y < 4$?

1. $(1, 1)$
2. $(2, 5)$
3. $(3, 2)$

A. I only

B. II only

C. I and II only

D. I and III only

A team hosting an event to raise money for new uniforms plans to sell at least 140 tickets before this event and at least 220 tickets during this event to raise a total of at least $\$5{,}820$ from all tickets sold. The price of a ticket during this event is $\$3$ less than the price of a ticket before this event. Which inequality represents this situation, where $x$ is the price, in dollars, of a ticket sold during this event?

A. $140(x + 3) + 220x \le 5{,}820$

B. $140(x + 3) + 220x \ge 5{,}820$

C. $140(x - 3) + 220x \le 5{,}820$

D. $140(x - 3) + 220x \ge 5{,}820$

In a set of four consecutive odd integers, where the integers are ordered from least to greatest, the first integer is represented by $x$. The product of 12 and the fourth odd integer is at most 26 less than the sum of the first and third odd integers. Which inequality represents this situation?

A. $12(x + 6) \le x + (x + 4) - 26$

B. $12(x + 6) \ge 26 - (x + (x + 4))$

C. $12(x + 4) \le x + (x + 3) - 26$

D. $12(x + 4) \ge 26 - (x + (x + 3))$

A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least 10 packages, and the total weight of the packages can be at most 1,100 pounds. What is the maximum number of 120-pound packages that the helicopter can carry per trip?

A. 2

B. 4

C. 5

D. 6

During a portion of a flight, a small airplane's cruising speed varied between 150 miles per hour and 170 miles per hour. Which inequality best represents this situation, where $s$ is the cruising speed, in miles per hour, during this portion of the flight?

A. $s \le 20$

B. $s \le 150$

C. $s \le 170$

D. $150 \le s \le 170$

In North America, the standard width of a parking space is at least 7.5 feet and no more than 9.0 feet. A restaurant owner recently resurfaced the restaurant’s parking lot and wants to determine the number of parking spaces, $n$, in the parking lot that could be placed perpendicular to a curb that is 135 feet long, based on the standard width of a parking space. Which of the following describes all the possible values of $n$?

A. $18 \le n \le 135$

B. $7.5 \le n \le 9$

C. $15 \le n \le 135$

D. $15 \le n \le 18$

For a 3-week period in a town in Illinois, the lowest recorded temperature was 31 degrees Fahrenheit ($^{\circ}F$) and the highest recorded temperature was 67$^{\circ}F$. Which inequality is true for any recorded temperature $t$, in $^{\circ}F$, in this town for this 3-week period?

A. $t \ge 98$

B. $t \ge 67$

C. $31 \le t \le 67$

D. $t \le 31$

The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12, which inequality represents the possible lengths, $x$, of the third side of the triangle?

A. $x < 18$

B. $x > 18$

C. $6 < x < 18$

D. $x < 6 \text{ or } x > 18$

$11x + 14y \le 115$

Anthony will spend at most $115 to purchase $x$ small cheese pizzas and $y$ large cheese pizzas for a team dinner. The given inequality represents this situation. Which of the following is the best interpretation of $14y$ in this context?

A. The amount, in dollars, Anthony will spend on each large cheese pizza

B. The amount, in dollars, Anthony will spend on each small cheese pizza

C. The total amount, in dollars, Anthony will spend on large cheese pizzas

D. The total amount, in dollars, Anthony will spend on small cheese pizzas

Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Ken saves 90% of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $270 for the week?

A. 38

B. 33

C. 22

D. 16

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

A clothing store is having a sale on shirts and pants. During the sale, the cost of each shirt is $15 and the cost of each pair of pants is $25. Geoff can spend at most $120 at the store. If Geoff buys $s$ shirts and $p$ pairs of pants, which of the following must be true?

A. $15s + 25p \le 120$

B. $15s + 25p \ge 120$

C. $25s + 15p \le 120$

D. $25s + 15p \ge 120$

$2x - y > 883$

For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given inequality?

A.

B.

C.

D.

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

A particular botanist classifies a species of plant as tall if its typical height when fully grown is more than 100 centimeters. Each of the following inequalities represents the possible heights $h$, in centimeters, for a specific plant species when fully grown. Which inequality represents the possible heights $h$, in centimeters, for a tall plant species?

A. $106 < h < 158$

B. $80 < h < 100$

C. $42 < h < 87$

D. $17 < h < 85$