Two-step equations and inequalities Transitional Reasoning Download PDF

Questions

1. 1.

   A biomedical laboratory uses an autoclave to sterilize surgical instruments under controlled conditions. Starting from a baseline atmospheric pressure of 101 kilopascals (kPa), the chamber's heating element increases the internal pressure at a steady rate of 4.5 kPa per minute. To prevent triggering the emergency relief valve, the chamber pressure must remain strictly below 173 kPa. Which inequality models the heating time, $m$, in minutes, and what is the maximum whole number of full minutes the heating element can operate before the valve is triggered?

   • A) $4.5m + 101 \le 173$; maximum of 16 minutes
   • B) $4.5m + 101 < 173$; maximum of 16 minutes
   • C) $101m + 4.5 < 173$; maximum of 1 minute
   • D) $4.5m + 101 < 173$; maximum of 15 minutes
2. 2.

   In an autonomous robotics competition, each team begins the course with a baseline score of 250 points. Judges deduct 8 points for every minor boundary infraction committed during the run. To qualify for the championship tier, a team's final score must be strictly greater than 186 points. Which inequality models the number of boundary infractions, $x$, a team can commit, and what is the maximum number of infractions allowed to qualify?

   • A) $250 - 8x > 186$; maximum of 7 infractions
   • B) $250 - 8x \ge 186$; maximum of 8 infractions
   • C) $250 - 8x > 186$; maximum of 8 infractions
   • D) $8x - 250 > 186$; maximum of 55 infractions
3. 3.

   A community youth club's treasury account has an overdraft balance of -$85 due to unexpected equipment repair costs. To qualify for an upcoming regional event, the club must bring its account balance to at least$340. If the club raises funds by selling handmade tote bags with a net profit of $17 per bag, which inequality models the number of tote bags, $b$, they must sell to meet their requirement, and what is the minimum number of bags needed?

   • A) $-85 + 17b \ge 340$, which gives $b \ge 25$; minimum of 25 bags
   • B) $17b + 85 \ge 340$, which gives $b \ge 15$; minimum of 15 bags
   • C) $-85 + 17b \le 340$, which gives $b \le 25$; maximum of 25 bags
   • D) $17b + 85 \le 340$, which gives $b \le 15$; maximum of 15 bags
4. 4.

   A civil engineering team is draining an excavation pit that contains 1,540 cubic meters of groundwater after a storm. A heavy-duty pump removes water at a constant rate of 45 cubic meters per hour. The construction crew can safely enter the pit only when the volume of remaining water has been reduced to no more than 280 cubic meters. Which inequality models the pumping time, $h$, in hours, and what is the minimum duration the pump must operate?

   • A) $1,540 - 45h \ge 280$, which requires at least 28 hours
   • B) $1,540 + 45h \le 280$, which requires at most 34 hours
   • C) $1,540 - 45h \le 280$, which requires at most 28 hours
   • D) $1,540 - 45h \le 280$, which requires at least 28 hours
5. 5.

   An oceanographic research vessel carries a continuous spool of fiber-optic communication cable of unknown length, $L$ meters. Technicians cut the cable into 8 equal sections to connect separate seafloor monitoring stations to a central data hub. Before installation, an additional 35 meters of reinforced protective shielding is spliced onto each individual section. If the total length of each finished station cable is exactly 185 meters, which equation models the original length, $L$, in meters, and what was that length?

   • A) $\frac{L + 35}{8} = 185$; $L = 1,445$ meters
   • B) $\frac{L}{8} + 35 = 185$; $L = 1,200$ meters
   • C) $\frac{L}{8} - 35 = 185$; $L = 1,760$ meters
   • D) $\frac{L - 35}{8} = 185$; $L = 1,515$ meters
6. 6.

   A wildlife biologist conducts an annual census of nesting sea turtles along a protected coastline. The survey records 75 active nesting sites this season. According to the conservation records, this count is 15 more than $\frac{3}{4}$ of the number of nesting sites recorded five years ago. Which equation models the number of nesting sites recorded five years ago, $p$, and what was that count?

   • A) $\frac{3}{4}p - 15 = 75$; $p = 120$
   • B) $\frac{3}{4}p + 15 = 75$; $p = 80$
   • C) $\frac{3}{4}p + 15 = 75$; $p = 45$
   • D) $\frac{3}{4}(p + 15) = 75$; $p = 85$
7. 7.

   A logistics company prepares a standardized shipping crate that has a maximum allowable gross weight of 850 kg. The empty crate and its protective lining have a combined weight of 130 kg. A shipping clerk loads 18 identical engine components into the crate. Which inequality represents the allowable weight, $w$, in kilograms, of each component so that the crate does not exceed the weight limit, and what is the maximum weight each component can have?

   • A) $18w - 130 \le 850$; the maximum weight is approximately 54.4 kg
   • B) $18w + 130 \le 850$; the maximum weight is 40 kg
   • C) $18w + 130 \ge 850$; the minimum weight is 40 kg
   • D) $18(w + 130) \le 850$; the maximum weight is approximately 32.2 kg
8. 8.

   A community science club organizes a fundraiser to purchase a new astronomical telescope. A local technology company agrees to match their fundraising by tripling the net proceeds after a fixed administrative fee of $75 is subtracted from the total amount raised. The club requires the final matched funding to be at least$1,800 to afford the telescope and its mounting tripod. Which inequality models the total amount raised, $x$, in dollars, and what is the minimum total amount the club must raise?

   • A) $3x - 75 \ge 1800$; minimum amount is $625
   • B) $3(x - 75) \ge 1800$; minimum amount is $600
   • C) $3(x - 75) \ge 1800$; minimum amount is $675
   • D) $3(x + 75) \ge 1800$; minimum amount is $525
9. 9.

   A landscape architect is designing a rectangular wildflower reserve inside a botanical park. The width of the reserve is planned to be exactly 16 meters. To protect the flora, the entire perimeter must be enclosed with fencing using no more than 110 meters of material. Which inequality models the possible length, $L$, in meters of the wildflower reserve, and what is the maximum length the architect can choose?

   • A) $2L + 32 \le 110$; maximum length is 39 meters
   • B) $2L + 16 \le 110$; maximum length is 47 meters
   • C) $2L + 32 \ge 110$; minimum length is 39 meters
   • D) $L + 32 \le 110$; maximum length is 78 meters
10. 10.

    A sales representative earns a fixed base salary of $2,500 per month plus a 6$4,000. Which inequality models the total sales amount, $s$, in dollars required to meet this goal, and what is the minimum sales amount she must achieve?

    • A) $0.06s + 2{,}500 \ge 4{,}000$; minimum sales of $25,000
    • B) $0.06s + 2{,}500 \le 4{,}000$; maximum sales of $25,000
    • C) $0.6s + 2{,}500 \ge 4{,}000$; minimum sales of $2,500
    • D) $0.06(s + 2{,}500) \ge 4{,}000$; minimum sales of $64,167
11. 11.

    A robotic submarine exploring an underwater canyon starts its recorded descent at an elevation of -120 meters relative to sea level. The submarine descends deeper into the canyon at a constant rate of 15 meters per minute until it settles on a survey shelf at an elevation of -495 meters. Which equation models the descent time, $m$, in minutes from the starting depth to the shelf, and how many minutes does this descent take?

    • A) $-120 - 15m = -495$; $m = 41$ minutes
    • B) $-120 - 15m = -495$; $m = 25$ minutes
    • C) $-120 + 15m = -495$; $m = 25$ minutes
    • D) $-120 + 15m = -495$; $m = 41$ minutes
12. 12.

    A community theater incurs fixed operating expenses of $1,450 per performance for venue rental, lighting, and sound equipment. Tickets for the opening night musical are sold at a price of$22.50 each. The production directors set a financial goal that the net revenue (ticket sales minus operating expenses) must strictly exceed $1,700. Which inequality models the number of tickets, $t$, that must be sold, and what is the minimum whole number of tickets needed to meet this goal?

    • A) $22.5t + 1450 \ge 1700$; minimum of 12 tickets
    • B) $22.5t - 1450 \ge 1700$; minimum of 140 tickets
    • C) $22.5t - 1450 > 1700$; minimum of 141 tickets
    • D) $22.5t - 1450 > 1700$; minimum of 140 tickets
13. 13.

    A youth soccer coach buys 8 identical team jerseys from an online sports supplier. The company charges a flat personalization fee of $6.50 on each individual jersey to print a custom name and number. If the total bill for the order before tax is$228.00, which equation models the base price, $j$, of a single jersey in dollars, and what is its base price?

    • A) $8(j - 6.50) = 228$; base price is $35.00
    • B) $8(j + 6.50) = 228$; base price is $22.00
    • C) $8j + 6.50 = 228$; base price is $27.69
    • D) $8(j + 6.50) = 228$; base price is $28.50
14. 14.

    To earn a certification in a technical robotics course, Marcus must achieve a mean score of at least 85 points across 4 module assessments. On the first 3 assessments, Marcus earned scores of 82, 79, and 87 points. Which inequality models the score, $s$, Marcus must achieve on the fourth assessment to earn the certification, and what is the solution for $s$?

    • A) $\frac{248 + s}{4} \ge 85$, which gives $s \ge 92$
    • B) $\frac{248 + s}{4} \le 85$, which gives $s \le 92$
    • C) $\frac{248 + s}{3} \ge 85$, which gives $s \ge 7$
    • D) $\frac{248 + s}{4} \ge 85$, which gives $s \ge 23$
15. 15.

    A youth robotics club receives a total grant of $g$ dollars that is split equally among its 4 project teams. Team Alpha spends $145 of its allocated share on microcontrollers and must have at least$80 remaining to purchase necessary sensor kits. Which inequality models the total grant amount, $g$, and what is the minimum total grant amount the club must receive?

    • A) $\frac{g}{4} - 145 \ge 80$; the minimum total grant amount is $900.
    • B) $\frac{g - 145}{4} \ge 80$; the minimum total grant amount is $465.
    • C) $\frac{g}{4} - 145 \ge 80$; the minimum total grant amount is $260.
    • D) $\frac{g}{4} + 145 \ge 80$; the minimum total grant amount is $320.
16. 16.

    An environmental scientist monitors the concentration of a dissolved mineral in an experimental water basin. The basin contains a baseline concentration of 14 milligrams per liter (mg/L), and an automated system infuses the mineral at a constant rate of 3.5 mg/L per hour. The infusion process stops as soon as the concentration reaches a target level of exactly 49 mg/L. Which equation models the infusion time, $h$, in hours, and what is the total duration of the process?

    • A) $14h + 3.5 = 49;\; h = 3.25\text{ hours}$
    • B) $3.5h - 14 = 49;\; h = 18\text{ hours}$
    • C) $3.5h + 14 = 49;\; h = 10\text{ hours}$
    • D) $3.5h + 14 = 49;\; h = 18\text{ hours}$
17. 17.

    A laboratory technician prepares a saline experiment using a flask containing an unknown volume, $V$, of liquid in milliliters. After 85 mL of liquid is discarded to remove sediment, the remaining volume is distributed equally among 6 separate testing beakers. Each beaker receives exactly 45 mL of the solution. Which equation models this situation, and what was the initial volume of liquid in the flask?

    • A) $\frac{V - 85}{6} = 45$; $V = 355$ mL
    • B) $\frac{V + 85}{6} = 45$; $V = 185$ mL
    • C) $\frac{V}{6} - 85 = 45$; $V = 780$ mL
    • D) $\frac{V - 85}{6} = 45$; $V = 270$ mL
18. 18.

    A mountain weather station records an initial air temperature of 4°C at sunset. Due to an incoming Arctic cold front, the temperature drops steadily by 2.5°C each hour. To prevent sensitive meteorological instruments from freezing damage, an automated heating system activates whenever the temperature falls strictly below -16°C. Which inequality represents the number of hours, $t$, after sunset until the heating system activates, and what is the solution for $t$?

    • A) $4 - 2.5t < -16$, which gives $t < 8$
    • B) $4 + 2.5t < -16$, which gives $t < -8$
    • C) $4 - 2.5t < -16$, which gives $t > 8$
    • D) $2.5t - 4 < -16$, which gives $t < -4.8$
19. 19.

    A cargo drone operates with a fully charged battery holding 92 watt-hours (Wh) of energy. The drone consumes energy at a steady rate of 6.5 Wh per kilometer of flight, and aviation safety rules require the flight to end with at least 20.5 Wh of reserve energy remaining. Which inequality models the possible flight distance $d$, in kilometers, and what is the maximum distance the drone can travel under these conditions?

    • A) $92 - 6.5d \ge 20.5$; maximum distance of 11 km
    • B) $92 - 6.5d \le 20.5$; maximum distance of 11 km
    • C) $6.5d - 92 \ge 20.5$; maximum distance of 17 km
    • D) $92 - 6.5d \ge 20.5$; maximum distance of 12 km
20. 20.

    A community garden project receives a grant of $460 to construct identical raised planting beds. A one-time delivery fee of$40 is charged for the bulk soil and materials, and each bed requires $35 in specialized lumber. If the project team must keep the total cost within the grant amount, what is the maximum number of raised planting beds they can construct?

    • A) 14 beds
    • B) 13 beds
    • C) 12 beds
    • D) 11 beds