One-variable equations and inequalities Transitional Reasoning Practice

Questions

1. 1.

   A mountain ski resort operates an aerial tramway cabin with an empty mass of 3,200 kg and a maximum certified gross weight of 7,000 kg. A cabin operator and essential safety gear with a combined mass of 200 kg are on board during every ascent. The cabin transports adult passengers averaging 70 kg each and child passengers averaging 40 kg each, maintaining twice as many adults as children on the trip. Which inequality models the allowable number of child passengers, $c$, and what is the maximum total number of passengers (adults and children combined) that can safely board the cabin?

   • A) $3400 + 110c \le 7000$; maximum of 32 passengers
   • B) $3400 + 180c \le 7000$; maximum of 20 passengers
   • C) $3200 + 180c \le 7000$; maximum of 63 passengers
   • D) $3400 + 180c \le 7000$; maximum of 60 passengers
2. 2.

   A municipal library receives a shipment of identical new reference volumes to arrange across a designated set of empty display shelves. If the librarians place 32 volumes on each shelf, 28 volumes remain without a shelf. If they instead place 40 volumes on each shelf, all shelves are completely filled except for the final shelf, which holds exactly 20 volumes. Which equation correctly models this situation in terms of the number of display shelves, $b$, and what is the total number of reference volumes in the shipment?

   • A) $32b + 28 = 40b + 20$; 60 volumes
   • B) $32b + 28 = 40(b - 1) + 20$; 192 volumes
   • C) $32(b - 1) + 28 = 40b - 20$; 256 volumes
   • D) $32b + 28 = 40(b - 1) + 20$; 220 volumes
3. 3.

   An oceanographic research station deploys an autonomous underwater glider to collect salinity data along a straight coastal corridor and return to base. Due to ocean currents, the glider travels outward at a constant speed of 15 km/h, but its return speed along the identical path is reduced to 10 km/h. To prevent battery depletion, the total round-trip journey must not exceed 6 hours. Which inequality models the allowable one-way distance in kilometers, $d$, and what is the maximum distance the glider can travel outward before returning?

   • A) $\frac{d}{15} + \frac{d}{10} \ge 6$; minimum distance is 36 kilometers
   • B) $\frac{2d}{25} \le 6$; maximum distance is 75 kilometers
   • C) $\frac{d}{15} + \frac{d}{10} \le 6$; maximum distance is 36 kilometers
   • D) $\frac{d}{15} + \frac{d}{10} \le 6$; maximum distance is 18 kilometers
4. 4.

   A custom furniture maker cuts pieces from a single 320-centimeter timber beam to build an open shelving unit. The design requires 4 identical upright supports of length $x$ centimeters and 3 identical crossbars that are each 12 centimeters shorter than an upright support. The cutting process produces a total saw kerf waste of 5 centimeters across all cuts, and workshop safety guidelines require that at least 15 centimeters of the beam must remain as an unused safety remnant. Which inequality models the allowable length, $x$, of an upright support, and what is the maximum possible whole-centimeter length of an upright support?

   • A) $4x + 3(x - 12) + 5 \le 320$; maximum length: 50 cm
   • B) $4x + 3(x + 12) + 20 \le 320$; maximum length: 37 cm
   • C) $4x + 3(x - 12) + 20 \le 320$; maximum length: 48 cm
   • D) $4x + 3(x - 12) + 20 \ge 320$; minimum length: 48 cm
5. 5.

   Two autonomous transport units depart simultaneously from opposite ends of a 420-meter straight logistics corridor, traveling toward each other along a designated track. Unit 1 travels at a constant speed of 1.8 meters per second, while Unit 2 travels at a constant speed of 1.2 meters per second. A safety system automatically activates warning beacons when the remaining distance between the two units narrows to 60 meters or less. Which inequality models the travel time in seconds, $t$, during which the warning beacons are activated, and what is the earliest time at which the beacons will turn on?

   • A) $420 - 3t \le 60$; earliest activation: 120 seconds
   • B) $420 - 3t \ge 60$; earliest activation: 120 seconds
   • C) $420 - 0.6t \le 60$; earliest activation: 600 seconds
   • D) $420 - 3t \le 60$; earliest activation: 160 seconds
6. 6.

   An oceanographic research vessel transmits environmental sensor data to an orbital satellite during scheduled transmission windows. A loss-free compression protocol reduces the size of the raw data by 35%, after which an uncompressed encryption header of 120 megabytes (MB) is attached to the payload. To complete transmission before the satellite passes out of range, the total uploaded payload must not exceed 900 MB. Which inequality models the allowable raw data size in megabytes, $d$, and what is the maximum raw data size that can be transmitted in a single pass?

   • A) $0.35d + 120 \le 900$; maximum raw data size is 2,229 MB
   • B) $0.65d + 120 \le 900$; maximum raw data size is 1,200 MB
   • C) $0.65d - 120 \le 900$; maximum raw data size is 1,569 MB
   • D) $1.35d + 120 \le 900$; maximum raw data size is 578 MB
7. 7.

   An electric vehicle charging station charges a flat connection fee of $4.50 plus$0.28 per kilowatt-hour (kWh) for the first 30 kWh, and a discounted rate of $0.20 per kWh for any energy consumed beyond 30 kWh. A driver has a maximum budget of$21.50 for a single charging session. If $x$ represents the additional kilowatt-hours consumed beyond the initial 30 kWh, which inequality models this situation, and what is the maximum total amount of electrical energy, in kWh, the driver can receive?

   • A) $12.90 + 0.20x \le 21.50$; maximum total energy of 43 kWh
   • B) $12.90 + 0.20x \le 21.50$; maximum total energy of 73 kWh
   • C) $4.50 + 0.20x \le 21.50$; maximum total energy of 85 kWh
   • D) $12.90 + 0.28x \le 21.50$; maximum total energy of 60 kWh
8. 8.

   A community youth foundation hosts an annual charity banquet to fund summer sports camps. A benefactor provides an upfront seed donation of $2,000 and pledges to contribute an additional$1.20 for every $1.00 raised from ticket sales. Each banquet ticket sells for$50, and the total operational expenses for venue rental and catering are fixed at 3,500. Which inequality correctly models the number of tickets, \(t\), the foundation must sell to secure a net profit of at least15,000 after paying all expenses, and what is the minimum number of tickets that must be sold?

   • A) $110t - 1500 \ge 15000$; minimum of 150 tickets
   • B) $110t + 2000 \ge 15000$; minimum of 119 tickets
   • C) $50t - 1500 \ge 15000$; minimum of 330 tickets
   • D) $60t - 1500 \ge 15000$; minimum of 275 tickets
9. 9.

   A municipal recreation department is designing a rectangular sports enclosure with a perimeter of at most 130 meters. The design specifications require the length of the enclosure to be 5 meters longer than twice its width. If $w$ represents the width of the enclosure in meters, which inequality represents all allowable values for the width, and what is the maximum allowable width?

   • A) $6w + 5 \le 130$; maximum width: 20.8 meters
   • B) $3w + 5 \le 130$; maximum width: 41.7 meters
   • C) $6w + 10 \le 130$; maximum width: 20 meters
   • D) $6w + 10 < 130$; maximum width: 19 meters
10. 10.

    An environmental research laboratory analyzes a 740-gram sediment core composed entirely of mineral silt, organic matter, and moisture. The mass of the mineral silt is twice the mass of the organic matter, and the moisture mass is 40 grams more than one-third the mass of the organic matter. Which equation correctly models the mass of the organic matter in grams, $m$, and what is the mass of the mineral silt in the core sample?

    • A) $\frac{10}{3}m + 40 = 740$; mineral silt: 210 g
    • B) $\frac{10}{3}m + 40 = 740$; mineral silt: 420 g
    • C) $\frac{7}{3}m + 40 = 740$; mineral silt: 600 g
    • D) $\frac{10}{3}m - 40 = 740$; mineral silt: 468 g
11. 11.

    A household energy storage system begins an evening period with 4.2 kWh of stored electricity. During the next 5 hours, rooftop solar panels supply an additional 0.6 kWh each hour to the system, while household appliances consume electricity at a constant rate of $r$ kWh each hour. To maintain emergency backup power, the stored electricity in the system must remain at or above 1.7 kWh throughout the entire 5-hour period. Which inequality models this condition, and what is the maximum allowable hourly consumption rate, $r$?

    • A) $4.2 + 5(0.6) - 5r \le 1.7$; maximum rate is 1.1 kWh/h
    • B) $4.2 - 5(0.6) + 5r \ge 1.7$; maximum rate is 0.5 kWh/h
    • C) $4.2 + 5(0.6) - 5r \ge 1.7$; maximum rate is 1.5 kWh/h
    • D) $4.2 + 5(0.6) - 5r \ge 1.7$; maximum rate is 1.1 kWh/h
12. 12.

    An artisan tea shop blends organic green tea priced at $12 per kilogram with premium jasmine blossoms priced at$20 per kilogram to create a 50-kilogram seasonal batch. To keep the retail price competitive, the total ingredient cost of the entire batch cannot exceed $760. If $x$ represents the mass in kilograms of organic green tea used in the batch, which inequality models this requirement, and what is the minimum mass of green tea that must be included?

    • A) $12x + 20(50 - x) \ge 760$; minimum mass: 30 kg
    • B) $12x + 20(50 - x) \le 760$; minimum mass: 30 kg
    • C) $12x + 20(50 - x) \le 760$; minimum mass: 20 kg
    • D) $20x + 12(50 - x) \le 760$; minimum mass: 25 kg
13. 13.

    An automated fulfillment center must pack a shipment of 1,020 identical gift sets using two packaging lines. Line A operates at a constant speed of 40 sets per hour, while Line B undergoes calibration and begins operating 3 hours after Line A at a constant speed of 60 sets per hour. Once started, both lines run simultaneously without interruption until the entire shipment is packed. Which equation correctly models this situation in terms of the total operating time of Line A in hours, $t$, and what is the total number of hours Line A operates?

    • A) $40(t - 3) + 60t = 1020$; Line A operates for 11.4 hours
    • B) $40t + 60(t + 3) = 1020$; Line A operates for 8.4 hours
    • C) $40t + 60(t - 3) = 1020$; Line A operates for 12 hours
    • D) $40t + 60(t - 3) = 1020$; Line A operates for 9 hours
14. 14.

    A conservation team monitors a reforested wetland parcel initially containing 1,450 native saplings, of which 20% are subsequently lost to an invasive blight. To qualify the parcel as an official ecological reserve, volunteers plan to plant $n$ disease-resistant saplings each month for 8 consecutive months so that the final tree count exceeds the original initial count by at least 150 saplings. Which inequality models the required monthly planting rate, $n$, and what is the minimum whole number of saplings that must be planted each month?

    • A) $1160 + 8n \ge 1600$; minimum of 55 saplings per month
    • B) $1450 + 8n \ge 1600$; minimum of 19 saplings per month
    • C) $1160 + 8n \ge 1450$; minimum of 37 saplings per month
    • D) $1160 + 8n \ge 1740$; minimum of 73 saplings per month
15. 15.

    A cooperative bakery prepares a daily batch of 50 artisan sourdough loaves with a fixed operating overhead of $120 and an ingredient cost of$2.50 per loaf. Each loaf sold brings in $7.50 in revenue, while every unsold loaf is donated to a community kitchen at an additional packaging and handling fee of$0.50 per donated loaf. On a day when the bakery achieved a target net profit of $90, all 50 loaves were either sold or donated. Which equation models the bakery's net profit in terms of the number of loaves sold, $x$, and how many loaves were sold that day?

    • A) $7.00x - 260 = 90$; $x = 50$ loaves
    • B) $7.50x - 240 = 90$; $x = 44$ loaves
    • C) $8.00x - 270 = 90$; $x = 45$ loaves
    • D) $8.00x - 230 = 90$; $x = 40$ loaves
16. 16.

    A student in a data science certification program has completed four module exams with scores of 78, 85, 92, and 81. The final capstone project counts as two exam grades toward the course average. To earn an honors distinction, the student's overall weighted average score must be at least 86. Which inequality models the capstone project score, $p$, needed to earn an honors distinction, and what is the minimum integer score the student must achieve?

    • A) $\frac{336 + p}{5} \ge 86$; minimum score: 94
    • B) $\frac{336 + 2p}{5} \ge 86$; minimum score: 47
    • C) $\frac{336 + 2p}{6} \ge 86$; minimum score: 90
    • D) $\frac{336 + 2p}{6} > 86$; minimum score: 91
17. 17.

    A community maker space orders supplies for a robotics workshop, purchasing 14 basic microcontroller boards and 6 advanced sensor expansion shields. Each advanced sensor expansion shield costs $25 more than a basic microcontroller board, and the distributor adds a flat shipping and handling fee of$22 to the entire order. If the total invoice amount is $572, which equation models the purchase in terms of the unit price of a basic microcontroller board, $x$, and what is the unit price of an advanced sensor expansion shield?

    • A) $14x + 6(x + 25) + 22 = 572$; unit price of an advanced shield: $20
    • B) $14(x + 25) + 6x + 22 = 572$; unit price of an advanced shield: $35
    • C) $14x + 6x + 25 + 22 = 572$; unit price of an advanced shield: $51.25
    • D) $14x + 6(x + 25) + 22 = 572$; unit price of an advanced shield: $45
18. 18.

    A delivery drone with an empty weight of 14.5 kg has a maximum permitted takeoff weight of 25.0 kg. The drone must carry a fixed navigation transmitter weighing 1.3 kg and is loaded with $k$ identical emergency supply kits, each weighing 0.8 kg. Flight regulations mandate that the total payload—consisting of the transmitter and all supply kits combined—must weigh at least 6.1 kg for aerodynamic balance, while the total takeoff weight must not exceed the maximum limit. Which compound inequality represents all allowable values for $k$, and what is the maximum whole number of kits the drone can transport?

    • A) $6.1 \le 1.3 + 0.8k$ and $15.8 + 0.8k \le 25.0$; maximum of 12 kits
    • B) $6.1 \le 0.8k$ and $14.5 + 0.8k \le 25.0$; maximum of 13 kits
    • C) $6.1 \le 1.3 + 0.8k$ and $14.5 + 0.8k \le 25.0$; maximum of 10 kits
    • D) $6.1 \le 1.3 + 0.8k$ and $15.8 + 0.8k \le 25.0$; maximum of 11 kits
19. 19.

    A refrigerated transport truck begins a cross-country route with an internal container temperature of $-18$°C. Due to a secondary cooling system failure, the internal temperature begins rising at a constant rate of $1.5$°C per hour. To prevent delicate dairy cargo from spoiling, safety protocols mandate that the temperature must remain strictly below $-3$°C. If $t$ represents the elapsed time in hours following the failure, which inequality expresses the permissible travel times to keep the cargo safe, and what is the maximum whole number of hours the driver can continue driving before the temperature threshold is breached?

    • A) $t < 10$ hours; maximum whole number of hours is 9
    • B) $t \le 10$ hours; maximum whole number of hours is 10
    • C) $t < 14$ hours; maximum whole number of hours is 13
    • D) $t \le 14$ hours; maximum whole number of hours is 14
20. 20.

    A city bike-share program offers two monthly membership options: Plan X charges a fixed fee of $18 plus$0.15 per minute of riding, while Plan Y charges a fixed fee of $6 plus$0.35 per minute of riding. A commuter wishes to choose Plan X only if it costs strictly less than Plan Y for their total monthly riding time. Which inequality represents all possible total monthly riding times in minutes, $m$, for which Plan X is strictly more economical than Plan Y?

    • A) $m < 60$
    • B) $m > 60$
    • C) $m > 48$
    • D) $m < 120$

Answers and step-by-step solutions appear after you finish the test.