One-variable equations and inequalities Transitional Reasoning Download PDF

Questions

1. 1.

   An ethical investment trust allocates a total portfolio of $15,000 between two community loan funds. Fund A offers a 6$1,170. If $x$ represents the amount of money in dollars allocated to Fund A, which inequality models this requirement, and what is the maximum amount the trust can allocate to Fund A?

   • A) $0.06x + 0.09(15,000 - x) \ge 1,170$; maximum allocation of $6,000
   • B) $0.06x + 0.09(15,000 - x) \ge 1,170$; minimum allocation of $6,000
   • C) $0.09x + 0.06(15,000 - x) \ge 1,170$; minimum allocation of $9,000
   • D) $0.06x + 0.09(15,000 - x) \ge 1,170$; maximum allocation of $9,000
2. 2.

   A robotics club is organizing its inventory of identical microcontrollers into storage bins. If the team places 12 microcontrollers in each bin, 26 microcontrollers remain left over without a bin. If they instead place 16 microcontrollers in each bin, all microcontrollers are accommodated, and the final bin has 6 empty slots while all other bins are completely full. Which linear equation models this situation in terms of the number of bins, $b$, and what is the total number of microcontrollers in the inventory?

   • A) $12b + 26 = 16b + 6$; total: 86 microcontrollers
   • B) $12b + 26 = 16(b - 1) + 6$; total: 134 microcontrollers
   • C) $12(b - 1) + 26 = 16b - 6$; total: 74 microcontrollers
   • D) $12b + 26 = 16b - 6$; total: 122 microcontrollers
3. 3.

   An agricultural cooperative packages organic walnuts into two gift crate sizes: a standard crate and a family crate. A standard crate contains 4 bags of raw walnuts and 2 bags of roasted walnuts, while a family crate contains 7 bags of raw walnuts and 5 bags of roasted walnuts. Each bag of roasted walnuts costs $3 more than a bag of raw walnuts, and the total production cost of a family crate is$33 more than that of a standard crate. If $r$ represents the cost in dollars of a single bag of raw walnuts, which equation correctly models this relationship, and what is the total production cost of a family crate?

   • A) Equation: $7r + 5(r - 3) = 4r + 2(r - 3) + 33$; Cost of a family crate: $69
   • B) Equation: $7r + 5(r + 3) = 4r + 2(r + 3) + 33$; Cost of a family crate: $63
   • C) Equation: $7r + 5(r + 3) = 4r + 2(r + 3) + 33$; Cost of a family crate: $30
   • D) Equation: $7r + 5(r + 3) = 4r + 2(r + 3) - 33$; Cost of a family crate: $45
4. 4.

   A conservation organization launches a fundraising campaign to restore a degraded wetland habitat. An environmental foundation pledges a challenge grant: once individual public donations surpass an initial threshold of $2,500, the foundation will donate$1.50 for every dollar contributed by the public beyond that threshold. The campaign aims to collect a combined total of at least $14,000 from public donations and foundation matching funds together. If $d$ represents the total amount of public donations in dollars, which inequality models this requirement, and what is the minimum amount the public must donate to reach the goal?

   • A) $d + 1.50d - 2{,}500 \ge 14{,}000$; minimum of $6,600
   • B) $1.50(d - 2{,}500) \ge 14{,}000$; minimum of $11,834
   • C) $d + 1.50(d + 2{,}500) \ge 14{,}000$; minimum of $4,100
   • D) $d + 1.50(d - 2{,}500) \ge 14{,}000$; minimum of $7,100
5. 5.

   An oceanographic research submersible begins a mission at an initial depth of 140 meters below the ocean surface and descends at a constant rate of 15 meters per minute for 20 minutes. After completing its observations, the submersible ascends toward the surface at a constant rate of 20 meters per minute for $m$ minutes. To safely establish a high-frequency communication link with its surface vessel, the submersible must reach a depth of at most 80 meters below the surface. Which inequality correctly models this condition in terms of the ascent time $m$, and what is the minimum number of minutes the submersible must ascend?

   • A) $440 - 20m \ge 80$, and the submersible must ascend for at least 18 minutes.
   • B) $440 - 20m \le 80$, and the submersible must ascend for at most 18 minutes.
   • C) $440 - 20m \le 80$, and the submersible must ascend for at least 18 minutes.
   • D) $300 - 20m \le 80$, and the submersible must ascend for at least 11 minutes.
6. 6.

   An environmental field technician earns a base hourly rate for the first 35 regular hours worked in a week and 1.5 times the base rate for each overtime hour beyond 35 hours. During a project week, the technician works 46 hours and has a mandatory equipment maintenance fee of $38 deducted from the gross earnings. If the technician receives a net pay of$1,198 for the week, which equation models the base hourly rate, $r$, and what is the value of $r$?

   • A) $35r + 1.5r(46 - 35) - 38 = 1{,}198$; $r = 24$
   • B) $35r + 1.5r(46 - 35) + 38 = 1{,}198$; $r = 22.52$
   • C) $46r + 1.5r(46 - 35) - 38 = 1{,}198$; $r = 19.78$
   • D) $35r + (46 - 35)r - 38 = 1{,}198$; $r = 26.87$
7. 7.

   Two wildlife researchers, Maya and Liam, conduct a biodiversity survey along a single 31-kilometer ecological corridor connecting two field stations. Maya departs from the northern station at 07:00 and travels south at a steady speed of 4 km/h. Liam departs from the southern station at 08:00 and travels north toward Maya at a steady speed of 5 km/h along the same route. If $t$ represents Maya's total travel time in hours until they meet, which equation correctly models this situation, and after how many hours from Maya's departure will the two researchers meet?

   • A) $4t + 5(t + 1) = 31$; they meet 3 hours after Maya's departure.
   • B) $4(t - 1) + 5t = 31$; they meet 4 hours after Maya's departure.
   • C) $4t + 5(t - 1) = 31$; they meet 3 hours after Maya's departure.
   • D) $4t + 5(t - 1) = 31$; they meet 4 hours after Maya's departure.
8. 8.

   A laboratory technician needs to prepare a total of 500 milliliters of a disinfectant solution with an active agent concentration between 24% and 30%, inclusive. To achieve this, the technician combines $x$ milliliters of a 15% disinfectant solution with $(500 - x)$ milliliters of a 40% disinfectant solution. Which compound inequality models this concentration requirement in terms of $x$, and what is the permissible range of volume for the 15% solution?

   • A) $120 \le 0.15x + 0.40(500 - x) \le 150$, which gives $200 \le x \le 320$
   • B) $120 \le 0.15x + 0.40(500 - x) \le 150$, which gives $160 \le x \le 280$
   • C) $120 \le 0.40x + 0.15(500 - x) \le 150$, which gives $180 \le x \le 300$
   • D) $24 \le 0.15x + 0.40(500 - x) \le 30$, which gives $680 \le x \le 704$
9. 9.

   An urban landscape architect is designing a rectangular botanical enclosure whose length is 6 meters greater than its width, $w$. To comply with site perimeter fencing limits, the enclosure's total perimeter cannot exceed 80 meters, while botanical standards require its length to be at least 18 meters. Which compound condition models these requirements in terms of width $w$, and what is the full range of permissible values for $w$?

   • A) $4w + 12 \le 80$ and $w + 6 \ge 18$, which yields $6 \le w \le 17$
   • B) $4w + 12 \le 80$ and $w + 6 \ge 18$, which yields $12 \le w \le 17$
   • C) $2w + 12 \le 80$ and $w + 6 \ge 18$, which yields $12 \le w \le 34$
   • D) $4w + 24 \le 80$ and $w + 6 \ge 18$, which yields $12 \le w \le 14$
10. 10.

    A civil engineering team is contracted to install a municipal drainage pipeline with a total length of at least 1,280 meters. During the initial phase, difficult ground conditions restrict the crew to installing only 85 meters per day for the first 4 days. To finish the project on or ahead of the contract deadline, the crew has 5 remaining days to complete the pipeline at a higher, constant daily pace. If $x$ represents the number of meters installed per day during the final 5 days, which inequality correctly models this requirement, and what is the minimum daily installation rate needed?

    • A) $340 + 5x \le 1280$, giving a maximum daily rate of 188 meters per day
    • B) $4(85 + x) \ge 1280$, giving a minimum daily rate of 235 meters per day
    • C) $340 + 5x \ge 1280$, giving a minimum daily rate of 188 meters per day
    • D) $5x - 340 \ge 1280$, giving a minimum daily rate of 324 meters per day
11. 11.

    A software quality analyst must achieve an average performance score of at least 88 across four equally weighted system evaluation audits to secure an advanced testing accreditation. The analyst has completed the first three audits with scores of 82, 91, and 85. If $x$ represents the score on the fourth audit, which inequality models this requirement, and what is the minimum score the analyst must achieve?

    • A) $\frac{82 + 91 + 85 + x}{4} \ge 88$; minimum score of 94
    • B) $\frac{82 + 91 + 85 + x}{4} \ge 88$; minimum score of 86
    • C) $\frac{82 + 91 + 85}{3} + x \ge 88$; minimum score of 90
    • D) $\frac{82 + 91 + 85 + x}{4} > 88$; minimum score of 95
12. 12.

    A commercial freight elevator has a maximum structural capacity of 1,500 kg. A technician weighing 82 kg is transporting a pallet jack weighing 118 kg along with several identical crates of hardware components, each having a mass of 65 kg. Facility safety standards require that the total weight inside the elevator during transport must not exceed 85% of its structural capacity. If $n$ represents the number of crates loaded, which inequality models this safety constraint, and what is the maximum number of complete crates that can be transported in one trip?

    • A) $65n + 200 \le 1500$; maximum of 20 crates
    • B) $65n + 118 \le 1275$; maximum of 17 crates
    • C) $65n + 200 \le 1275$; maximum of 16 crates
    • D) $65n + 200 \ge 1275$; minimum of 17 crates
13. 13.

    A candidate takes a 50-question technical certification exam where each correct response earns 4 points, each incorrect response deducts 1.5 points, and all questions must be answered. To obtain a professional certificate, the candidate must achieve a total score of at least 145 points. If $c$ represents the number of correct responses, which inequality models this requirement, and what is the minimum number of correct responses needed to earn the certificate?

    • A) $4c - 1.5(50 - c) \le 145$; at least 40 correct answers
    • B) $4c - 1.5(50 - c) \ge 145$; at least 38 correct answers
    • C) $4c - 1.5(50 - c) \ge 145$; at least 40 correct answers
    • D) $4c - (50 - c) \ge 145$; at least 39 correct answers
14. 14.

    A structural engineering team is designing an 84-meter-long pedestrian walkway that consists of two identical outer approach sections and one central bridge span. The design specifications require the central bridge span to be 6 meters shorter than three times the length of a single outer approach section. Which linear equation correctly models this walkway layout in terms of the length of an outer approach section, $x$, and what is the actual length of the central bridge span?

    • A) $5x - 6 = 84$; central span is 48 meters
    • B) $5x - 6 = 84$; central span is 18 meters
    • C) $4x - 6 = 84$; central span is 61.5 meters
    • D) $5x + 6 = 84$; central span is 52.8 meters
15. 15.

    An aquatic center manages an Olympic training pool that currently has a chlorine disinfectant level of 1.20 parts per million (ppm). An automated dosing system increases the chlorine level by 0.35 ppm per hour, while ultraviolet breakdown and surface aeration reduce it by 0.10 ppm per hour. According to local public health regulations, the pool can only open when the chlorine level is at least 2.70 ppm, but it must not exceed 4.20 ppm to ensure swimmer safety. If $t$ represents the operating time in hours from this starting point, which compound inequality models the required chlorine concentration, and what is the complete range of operating hours during which the pool meets opening standards?

    • A) $2.70 \le 1.20 + 0.45t \le 4.20$, which simplifies to $3.33 \le t \le 6.67$
    • B) $2.70 \le 0.25t \le 4.20$, which simplifies to $10.8 \le t \le 16.8$
    • C) $2.70 \le 1.20 + 0.25t \le 4.20$, which simplifies to $6 \le t \le 16.8$
    • D) $2.70 \le 1.20 + 0.25t \le 4.20$, which simplifies to $6 \le t \le 12$
16. 16.

    A regional logistics center evaluates two transport providers for dispatching freight batches. Provider A charges a flat dispatch fee of $45 plus$3.20 per kilogram of cargo, whereas Provider B charges no flat fee but costs $4.70 per kilogram. Which inequality correctly represents the condition under which Provider A is strictly more economical than Provider B for a batch weighing $w$ kilograms, and what is the smallest whole number of kilograms that satisfies this condition?

    • A) $45 + 3.20w < 4.70w$; minimum 31 kg
    • B) $45 + 3.20w \le 4.70w$; minimum 30 kg
    • C) $45 + 4.70w < 3.20w$; minimum 30 kg
    • D) $45 + 3.20w > 4.70w$; minimum 31 kg
17. 17.

    A digital printing workshop produces custom promotional banners for corporate events. For each production run, the workshop incurs a fixed calibration cost of $85, plus a variable material cost of$14 per banner. Each banner is sold at a fixed price of 31. If \(b\) represents the number of banners printed and sold in a single production run, which inequality models the condition needed to earn a net profit of at least680, and what is the minimum number of banners required?

    • A) $17b - 85 \ge 680$; minimum of 45 banners
    • B) $17b + 85 \ge 680$; minimum of 35 banners
    • C) $17b \ge 680$; minimum of 40 banners
    • D) $45b - 85 \ge 680$; minimum of 17 banners
18. 18.

    A community greenhouse maintains a rainwater collection tank holding 1,200 liters of water at the start of a dry week. Daily automated irrigation uses 140 liters, while backwashing the filtration system discharges an additional 35 liters each day. To prevent sediment damage, the pump operates only as long as the water volume in the tank remains at or above 250 liters. Which inequality represents the condition for the pump to remain operational after $d$ full days, and what is the maximum number of full days it can run?

    • A) $1{,}200 - 175d \ge 250$; at most 5 full days
    • B) $1{,}200 - 175d \ge 250$; at most 6 full days
    • C) $1{,}200 - 105d \ge 250$; at most 9 full days
    • D) $1{,}200 + 175d \le 250$; at most 5 full days