Quick Answer
Budgets, weight restrictions, delivery costs, scheduling, minimum scores, and other real-world scenarios are transformed into single-variable inequalities and then solved in SAT linear inequality word questions. This page has55 SAT-style word problem questions, organised with complete solved answers with trap notes, ranging from simple “translate the sentence” tasks to multi-step, two-plan comparison problems.
What to Know Before You Start
- Word problems test translation first and algebra second — most students lose points on the setup, not the solving.
- Key phrases map directly to symbols: “at least” → ≥, “at most” / “no more than” → ≤, “more than” / “exceeds” → >, “fewer than” / “under” → <.
- Fixed costs (setup fees, base charges, flat rates) stay outside the variable term; variable costs (per-item, per-hour, per-mile) get multiplied by the unknown.
- Many “maximum number of ___” problems require rounding down to a whole number even if the algebra gives a decimal.
- “Compare two plans” problems usually reduce to a single linear inequality once you set Plan A’s expression against Plan B’s.
- Always re-read the question after solving — the algebra may ask for x, but the question may ask for a different related quantity.
In This Guide – 55 Word Problem Practice Questions
- What do SAT inequality word problems test?
- Budget & cost constraint problems
- Weight, capacity & quantity limit problems
- Minimum score, target & threshold problems
- Comparing two plans or rates
- Multi-step and mixed reasoning word problems
- Common mistakes on word problems
- 2-week study plan for word problems
- Frequently asked questions
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What Do SAT Inequality Word Problems Test?
The Digital SAT’s Algebra domain contains word problems that nearly always depict a real-world limitation, such as a budget that cannot be exceeded, a container that cannot be overfilled, a score that must be attained, or a schedule that must fit. The arithmetic is a linear inequality that can be solved in one or two steps. The translation step—accurately determining the fixed amount, the variable amount, and the direction of the inequality before writing anything down—is what distinguishes a quick, accurate solver from a slow one.
| Word Problem Type | What It Tests | Common Trap | Practice Set |
|---|---|---|---|
| Budget & cost | Fixed vs. variable cost setup | Swapping fixed and per-unit terms | Q1–Q13 |
| Weight / capacity / quantity | “No more than” limits, rounding down | Forgetting whole-number answers | Q14–Q26 |
| Minimum score / threshold | “At least” translation, rounding up | Reversing ≥ and ≤ | Q27–Q37 |
| Comparing two plans/rates | Setting up one inequality from two expressions | Misreading which plan is “more” | Q38–Q47 |
| Multi-step / mixed | Combining constraints, multi-part reasoning | Solving before fully reading the constraint | Q48–Q55 |
SAT strategy:Prior to constructing an inequality, highlight the fixed amount, the amount per unit, and the limit phrase (“at least,” “at most,” “more than,” “fewer than”). translate to symbols only after that.
Budget & Cost Constraint Word Problems
In these challenges, a budget ceiling or floor is combined with a fixed cost and a cost per unit.
Budget & Cost · Easy · Question 1
A caterer charges $15 per guest in addition to a $120 fixed cost. The maximum budget for a client is $600. Which inequality yields the maximum number of visitors (g)?
A) 120 + 15g ≤ 600
B) 15 + 120g ≤ 600
C) 120 + 15g ≥ 600
D) 120g + 15 ≤ 600
Show full solution
The flat fee ($120) is fixed; the per-guest charge ($15) multiplies g. “At most $600” means the total is ≤ 600. Answer: A
Budget & Cost · Medium · Question 2
Using the caterer from Question 1, what is the maximum number of guests the client can invite?
A) 30
B) 32
C) 34
D) 40
Show full solution
120 + 15g ≤ 600 → 15g ≤ 480 → g ≤ 32. Since g must be a whole number, the maximum is 32. Answer: B
Budget & Cost · Easy · Question 3
In addition to a $25 setup fee, a streaming service costs $9 per month. A student’s goal for the first year is to spend less than $150. Which inequality determines how many months the student can sign up for?
A) 25 + 9m < 150
B) 9 + 25m < 150
C) 25 + 9m ≤ 150
D) 9m < 150
Show full solution
The $25 setup fee is fixed; $9 per month multiplies m. “Less than $150” means strict <. Answer: A
Budget & Cost · Medium · Question 4
In addition to a $4 binding cost, a print shop charges $0.10 per page. A client has sixteen dollars to spend. What is the maximum number of pages that can be printed?
A) 100
B) 110
C) 120
D) 160
Show full solution
4 + 0.10p ≤ 16 → 0.10p ≤ 12 → p ≤ 120. Answer: C
Budget & Cost · Easy · Question 5
The base rate for a ride-sharing trip is $2.50 plus $1.75 each mile. A rider’s travel spending is limited at $20. The possible number of miles r is given by which inequality?
A) 2.50 + 1.75r ≤ 20
B) 1.75 + 2.50r ≤ 20
C) 2.50 + 1.75r ≥ 20
D) 2.50r ≤ 20
Show full solution
Base fare is fixed; per-mile rate multiplies r. “No more than” means ≤. Answer: A
Budget & Cost · Medium · Question 6
What is the longest distance a rider may travel while staying within their budget using the ride-share from Question 5??
A) 9
B) 10
C) 11
D) 12
Show full solution
1.75r ≤ 17.5 → r ≤ 10. Answer: B
Budget & Cost · Medium · Question 7
A gym charges a $60 enrollment fee plus $35 per month. A member wants total spending to exceed $200 after how many months at minimum?
A) 4
B) 5
C) 6
D) 3
Show full solution
60 + 35m > 200 → 35m > 140 → m > 4, so the minimum whole number of months is 5. Answer: B
Budget & Cost · Hard · Question 8
A wedding venue charges $2,000 plus $85 per guest. The couple’s total budget must not exceed $6,500, and the venue requires at least 30 guests. Which range of guest counts n is possible?
A) 30 ≤ n ≤ 52
B) 30 ≤ n ≤ 53
C) 30 ≤ n ≤ 76
D) 30 ≤ n < 52
Show full solution
2000 + 85n ≤ 6500 → 85n ≤ 4500 → n ≤ 52.9…, so n ≤ 52 as a whole number. Combined with n ≥ 30: 30 ≤ n ≤ 52. Answer: A
Budget & Cost · Easy · Question 9
A landscaping crew charges $75 plus $20 per hour. A homeowner has budgeted $250. Which inequality gives the possible number of hours h the crew can work?
A) 75 + 20h ≤ 250
B) 20 + 75h ≤ 250
C) 75h + 20 ≤ 250
D) 75 + 20h ≥ 250
Show full solution
Fixed $75, variable $20 per hour, budget cap means ≤. Answer: A
Budget & Cost · Medium · Question 10
Using the crew from Question 9, what is the maximum number of full hours the crew can work?
A) 7
B) 8
C) 8.75
D) 9
Show full solution
20h ≤ 175 → h ≤ 8.75. Since only full hours count, the maximum is 8. Answer: B
Budget & Cost · Medium · Question 11
A club is selling raffle tickets to raise more than $1,800 for a trip. Tickets cost $12 each, and the club has already collected $360 from sponsors. Which inequality gives the number of tickets t needed?
A) 360 + 12t > 1800
B) 12 + 360t > 1800
C) 360 + 12t ≥ 1800
D) 12t > 1800
Show full solution
Sponsor money is fixed; ticket sales are variable. “More than” means strict >. Answer: A
Budget & Cost · Hard · Question 12
Using the raffle from Question 11, what is the minimum number of tickets that must be sold?
A) 120
B) 121
C) 119
D) 180
Show full solution
12t > 1440 → t > 120. Since t must be a whole number greater than 120, the minimum is 121. Answer: B
Budget & Cost · Medium · Question 13
A phone plan costs $20 per month plus $0.05 per text over a limit. If a customer’s bill must stay under $35, which inequality gives the number of extra texts x allowed?
A) 20 + 0.05x < 35
B) 0.05 + 20x < 35
C) 20 + 0.05x ≤ 35
D) 0.05x < 35
Show full solution
Base plan is fixed; extra texts are variable. “Under $35” is strict <. Answer: A
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Weight, Capacity & Quantity Limit Word Problems
These problems involve a maximum capacity (weight, volume, or count) that cannot be exceeded, usually combined with a fixed starting amount.
Weight & Capacity · Easy · Question 14
An elevator has a maximum weight capacity of 2,000 pounds. Four people already weighing a combined 620 pounds are inside. Each additional box weighs 45 pounds. Which inequality gives the possible number of boxes b?
A) 620 + 45b ≤ 2000
B) 45 + 620b ≤ 2000
C) 620 + 45b ≥ 2000
D) 45b ≤ 2000
Show full solution
People’s weight is fixed; box weight is variable. Capacity limit means ≤. Answer: A
Weight & Capacity · Medium · Question 15
Using the elevator from Question 14, what is the maximum number of boxes that can be added?
A) 30
B) 31
C) 32
D) 33
Show full solution
45b ≤ 1380 → b ≤ 30.67, so the whole-number maximum is 30. Answer: A
Weight & Capacity · Easy · Question 16
A water tank holds at most 500 gallons. It currently has 140 gallons. Which inequality gives the amount of water w that can still be added?
A) w ≤ 360
B) w ≥ 360
C) w ≤ 500
D) w ≤ 640
Show full solution
140 + w ≤ 500 → w ≤ 360. Answer: A
Weight & Capacity · Medium · Question 17
A shipping container can carry no more than 1,200 kilograms. Each crate weighs 28 kilograms, and the pallet itself weighs 60 kilograms. What is the maximum number of crates c that can be loaded?
A) 40
B) 41
C) 42
D) 45
Show full solution
60 + 28c ≤ 1200 → 28c ≤ 1140 → c ≤ 40.7, so the maximum whole number is 40. Answer: A
Weight & Capacity · Easy · Question 18
An auditorium seats no more than 340 people. If 118 seats are already reserved for staff, which inequality gives the number of additional tickets t that can be sold?
A) t ≤ 222
B) t ≥ 222
C) t ≤ 340
D) t ≤ 458
Show full solution
118 + t ≤ 340 → t ≤ 222. Answer: A
Weight & Capacity · Medium · Question 19
A food truck’s cooler can hold at most 80 pounds of ice. Each bag of ice weighs 7 pounds, and the truck starts each day with 12 pounds already in the cooler. Which inequality gives the number of bags n the crew can add?
A) 12 + 7n ≤ 80
B) 7 + 12n ≤ 80
C) 12 + 7n ≥ 80
D) 7n ≤ 80
Show full solution
Starting ice is fixed; bags added are variable. Capacity means ≤. Answer: A
Weight & Capacity · Hard · Question 20
Using the food truck from Question 19, what is the maximum number of whole bags that can be added?
A) 9
B) 10
C) 11
D) 12
Show full solution
7n ≤ 68 → n ≤ 9.7, so the maximum whole number is 9. Answer: A
Weight & Capacity · Medium · Question 21
A moving van’s cargo area has a volume limit that fits at most 60 identical boxes. If 17 boxes are already loaded, which inequality gives the number of additional boxes x that can still fit?
A) x ≤ 43
B) x ≥ 43
C) x ≤ 60
D) x ≤ 77
Show full solution
17 + x ≤ 60 → x ≤ 43. Answer: A
Weight & Capacity · Hard · Question 22
A bridge sign restricts trucks to no more than 8,000 pounds. An empty truck weighs 5,200 pounds, and each identical pallet weighs 175 pounds. How many pallets can the truck carry at most?
A) 15
B) 16
C) 17
D) 18
Show full solution
5200 + 175p ≤ 8000 → 175p ≤ 2800 → p ≤ 16. Answer: B
Weight & Capacity · Easy · Question 23
A parking lot has space for no more than 150 cars. Currently 95 cars are parked. Which inequality gives the number of additional cars c that can park?
A) c ≤ 55
B) c ≥ 55
C) c ≤ 150
D) c ≤ 245
Show full solution
95 + c ≤ 150 → c ≤ 55. Answer: A
Weight & Capacity · Medium · Question 24
A hiking backpack should carry no more than 35 pounds. The empty pack weighs 4 pounds, and each water bottle weighs 2.5 pounds. How many bottles can be added at most, if 12 pounds of other gear is also packed?
A) 6
B) 7
C) 8
D) 9
Show full solution
4 + 12 + 2.5n ≤ 35 → 2.5n ≤ 19 → n ≤ 7.6, so the maximum whole number is 7. Answer: B
Weight & Capacity · Medium · Question 25
A conference room fits at most 48 chairs. Organizers want to leave at least 6 chairs unused for accessibility. Which inequality gives the number of attendees a who can be seated?
A) a ≤ 42
B) a ≥ 42
C) a ≤ 48
D) a ≤ 54
Show full solution
a + 6 ≤ 48 → a ≤ 42. Answer: A
Weight & Capacity · Hard · Question 26
A cargo elevator has a weight limit of 3,600 pounds and a passenger limit of 12 people. If the average passenger weighs 165 pounds and each carries a bag weighing at most 20 pounds, is a full load of 12 people with bags guaranteed to be under the weight limit?
A) Yes, since 12(165 + 20) = 2220 < 3600
B) No
C) Only if bags weigh less than 10 pounds
D) Cannot be determined
Show full solution
Maximum possible total weight is 12 × (165 + 20) = 12 × 185 = 2220 pounds, which is less than 3600. Answer: A
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Minimum Score, Target & Threshold Word Problems
These problems require reaching or exceeding a target — a test average, a sales goal, or a fundraising total.
Threshold · Easy · Question 27
A student needs a combined score of at least 380 points on two tests. She scored 172 on the first test. Which inequality gives the possible scores s on the second test?
A) s ≥ 208
B) s ≤ 208
C) s ≥ 552
D) s ≥ 380
Show full solution
172 + s ≥ 380 → s ≥ 208. Answer: A
Threshold · Medium · Question 28
A salesperson earns $450 per week plus $30 per sale. To earn more than $750 in a week, what is the minimum number of sales needed?
A) 10
B) 11
C) 9
D) 12
Show full solution
450 + 30s > 750 → 30s > 300 → s > 10, so the minimum whole number is 11. Answer: B
Threshold · Easy · Question 29
A charity needs to raise at least $3,200. So far $1,100 has been raised, and each additional donor gives $75. Which inequality gives the minimum number of additional donors d needed?
A) d ≥ 28
B) d ≥ 29
C) d ≥ 30
D) d ≥ 42
Show full solution
1100 + 75d ≥ 3200 → 75d ≥ 2100 → d ≥ 28. Answer: A
Threshold · Medium · Question 30
A student has read 60 pages of a 320-page book and plans to read 30 pages per day. What is the minimum number of days needed to have read at least 270 pages total?
A) 6
B) 7
C) 8
D) 9
Show full solution
60 + 30d ≥ 270 → 30d ≥ 210 → d ≥ 7. Answer: B
Threshold · Medium · Question 31
A factory must produce at least 500 units per day but has capacity for at most 800 units. Which inequality represents the possible daily production p?
A) 500 ≤ p ≤ 800
B) 500 < p < 800
C) p ≤ 500 or p ≥ 800
D) p ≥ 1300
Show full solution
“At least” includes the boundary (≥), and “at most” includes the boundary (≤), giving a closed compound inequality. Answer: A
Threshold · Hard · Question 32
A student’s average on 4 tests must be at least 88 to earn an A. Her first three scores are 85, 90, and 84. What is the minimum score needed on the fourth test?
A) 93
B) 94
C) 95
D) 92
Show full solution
(85 + 90 + 84 + s)/4 ≥ 88 → 259 + s ≥ 352 → s ≥ 93. Answer: A
Threshold · Easy · Question 33
A recipe requires the oven temperature to stay between 300°F and 350°F, inclusive. Which inequality represents the allowed temperature T?
A) 300 ≤ T ≤ 350
B) 300 < T < 350
C) T ≤ 300 or T ≥ 350
D) 300 ≤ T < 350
Show full solution
“Inclusive” means both endpoints are included, so both symbols are ≤. Answer: A
Threshold · Medium · Question 34
A runner wants a total distance of at least 40 miles this week. She has already run 14 miles and plans to run 3.5 miles per session for the remaining days. What is the minimum number of sessions needed?
A) 7
B) 8
C) 9
D) 6
Show full solution
14 + 3.5s ≥ 40 → 3.5s ≥ 26 → s ≥ 7.43, so the minimum whole number of sessions is 8. Answer: B
Threshold · Medium · Question 35
A vending machine business needs weekly profit to exceed $900. Each item sells for $2.25, and fixed weekly costs are $360. What is the minimum number of items that must sell?
A) 400
B) 401
C) 560
D) 561
Show full solution
Profit = revenue − costs, so 2.25n − 360 > 900 → 2.25n > 1260 → n > 560, and the minimum whole number is 561. Answer: D
Threshold · Easy · Question 36
A cyclist wants to average at least 15 miles per hour over a 2-hour ride. Which inequality gives the total distance d (in miles) needed?
A) d ≥ 30
B) d ≤ 30
C) d ≥ 15
D) d ≥ 7.5
Show full solution
Average speed × time = distance: 15 × 2 = 30, so d ≥ 30. Answer: A
Threshold · Hard · Question 37
A company’s profit P (in dollars) is modeled by P = 40u − 900, where u is units sold. What is the minimum number of units needed for the company to be profitable (P > 0)?
A) 22
B) 23
C) 21
D) 24
Show full solution
40u − 900 > 0 → 40u > 900 → u > 22.5, so the minimum whole number is 23. Answer: B
Comparing Two Plans or Rates
These problems set two linear expressions against each other and ask when one option becomes cheaper, faster, or larger than the other.
Comparing Plans · Medium · Question 38
Gym A charges $25 per month with no sign-up fee. Gym B charges $15 per month plus a $70 sign-up fee. After how many months does Gym B become cheaper than Gym A?
A) more than 7 months
B) more than 6 months
C) fewer than 7 months
D) more than 8 months
Show full solution
Gym B is cheaper when 15m + 70 < 25m → 70 < 10m → m > 7. Answer: A
Comparing Plans · Medium · Question 39
Plan A costs $12 plus $0.06 per minute. Plan B costs $30 with unlimited minutes. For how many minutes does Plan A cost more than Plan B?
A) more than 300 minutes
B) more than 280 minutes
C) more than 320 minutes
D) more than 250 minutes
Show full solution
12 + 0.06t > 30 → 0.06t > 18 → t > 300. Answer: A
Comparing Plans · Hard · Question 40
Rental Company A charges $40 per day plus $0.20 per mile. Rental Company B charges $55 per day with unlimited mileage. For a one-day rental, how many miles must be driven for Company B to be the cheaper option?
A) more than 75 miles
B) more than 65 miles
C) more than 80 miles
D) fewer than 75 miles
Show full solution
Company B is cheaper when 55 < 40 + 0.20x → 15 < 0.20x → x > 75. Answer: A
Comparing Plans · Medium · Question 41
Painter A charges $200 plus $18 per hour. Painter B charges a flat $350 regardless of hours. For how many hours of work is Painter A the cheaper choice?
A) fewer than about 8.3 hours
B) more than about 8.3 hours
C) fewer than 10 hours
D) more than 10 hours
Show full solution
200 + 18h < 350 → 18h < 150 → h < 8.33. Answer: A
Comparing Plans · Hard · Question 42
Internet Provider A charges $45 per month with a $99 installation fee. Provider B charges $60 per month with no installation fee. After how many full months does Provider A become the cheaper total cost?
A) after 7 months
B) after 6 months
C) after 8 months
D) after 5 months
Show full solution
99 + 45m < 60m → 99 < 15m → m > 6.6, so Provider A becomes cheaper starting in month 7. Answer: A
Comparing Plans · Medium · Question 43
A parking garage offers two options: pay $4 per hour, or buy a monthly pass for $80 with unlimited parking. For how many hours of parking per month does the monthly pass become cheaper?
A) more than 20 hours
B) more than 16 hours
C) fewer than 20 hours
D) more than 24 hours
Show full solution
80 < 4h → h > 20. Answer: A
Comparing Plans · Hard · Question 44
Two delivery services charge as follows: Service X costs $6 plus $1.50 per package; Service Y costs $2 plus $2.25 per package. For what number of packages does Service X cost less than Service Y?
A) more than about 5.3 packages
B) fewer than about 5.3 packages
C) more than 6 packages
D) fewer than 4 packages
Show full solution
6 + 1.50p < 2 + 2.25p → 4 < 0.75p → p > 5.33. Answer: A
Comparing Plans · Medium · Question 45
A tutoring center offers a single-session rate of $60 or a package of 5 sessions for $250. Beyond how many sessions does the package rate become the better deal per session?
A) more than about 4.2 sessions
B) more than 5 sessions
C) fewer than 5 sessions
D) more than 6 sessions
Show full solution
250 < 60s → s > 4.17, so from the 5th session onward the package is the better value. Answer: A
Comparing Plans · Hard · Question 46
Two savings plans start at different amounts: Plan M starts with $500 and adds $40 per week; Plan N starts with $260 and adds $65 per week. After how many weeks does Plan N have more money than Plan M?
A) more than about 9.6 weeks
B) fewer than about 9.6 weeks
C) more than 12 weeks
D) more than 8 weeks
Show full solution
260 + 65w > 500 + 40w → 25w > 240 → w > 9.6. Answer: A
Comparing Plans · Medium · Question 47
A print shop offers Option 1: $0.15 per flyer, or Option 2: $25 flat fee for up to 300 flyers. For how many flyers is Option 1 cheaper?
A) fewer than about 167 flyers
B) more than about 167 flyers
C) fewer than 150 flyers
D) more than 200 flyers
Show full solution
0.15f < 25 → f < 166.67, so Option 1 is cheaper for fewer than about 167 flyers. Answer: A
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Multi-Step & Mixed Reasoning Word Problems
These combine two constraints, require reading carefully for what quantity is actually being asked for, or involve compound conditions.
Multi-Step · Hard · Question 48
A concert venue holds between 500 and 800 people, inclusive. Tickets cost $45 each. Which compound inequality gives the possible revenue R from a sold-out show?
A) 22,500 ≤ R ≤ 36,000
B) 22,500 < R < 36,000
C) 500 ≤ R ≤ 800
D) 45 ≤ R ≤ 36,000
Show full solution
Multiply each part of 500 ≤ n ≤ 800 by 45: 22,500 ≤ R ≤ 36,000. Answer: A
Multi-Step · Hard · Question 49
A club needs at least 30 members but its meeting room fits at most 55 people. If 12 non-members also attend meetings, what is the range of possible member counts m?
A) 30 ≤ m ≤ 43
B) 30 ≤ m ≤ 55
C) 18 ≤ m ≤ 43
D) 30 ≤ m ≤ 67
Show full solution
Room capacity: m + 12 ≤ 55 → m ≤ 43. Combined with m ≥ 30: 30 ≤ m ≤ 43. Answer: A
Multi-Step · Hard · Question 50
A company’s weekly profit must stay between $2,000 and $5,000, inclusive, for a project to remain viable. Profit is modeled by P = 60u − 400, where u is units sold. What is the range of possible values of u?
A) 40 ≤ u ≤ 90
B) 33 ≤ u ≤ 83
C) 40 ≤ u ≤ 83
D) 33 ≤ u ≤ 90
Show full solution
2000 ≤ 60u − 400 ≤ 5000 → 2400 ≤ 60u ≤ 5400 → 40 ≤ u ≤ 90. Answer: A
Multi-Step · Hard · Question 51
A delivery driver must complete at least 18 deliveries but no more than 25 in a shift. Each delivery earns $12, and the driver also receives a flat $40 shift bonus. What is the range of possible total earnings E?
A) $256 ≤ E ≤ $340
B) $216 ≤ E ≤ $300
C) $18 ≤ E ≤ $25
D) $256 ≤ E ≤ $300
Show full solution
E = 40 + 12n where 18 ≤ n ≤ 25. Minimum: 40 + 12(18) = 256. Maximum: 40 + 12(25) = 340. Answer: A
Multi-Step · Hard · Question 52
A student’s final grade requires the average of 5 quizzes to be at least 80. Her first four scores are 76, 82, 79, and 85. To guarantee passing, which inequality must the fifth score q satisfy?
A) q ≥ 78
B) q ≥ 80
C) q ≥ 82
D) q ≥ 76
Show full solution
(76 + 82 + 79 + 85 + q)/5 ≥ 80 → 322 + q ≥ 400 → q ≥ 78. Answer: A
Multi-Step · Hard · Question 53
A theater sells adult tickets for $18 and student tickets for $10. On one night, exactly 40 student tickets were sold, and total revenue must be at least $1,000. Which inequality gives the minimum number of adult tickets a needed?
A) a ≥ 34
B) a ≥ 33
C) a ≥ 35
D) a ≥ 25
Show full solution
18a + 10(40) ≥ 1000 → 18a + 400 ≥ 1000 → 18a ≥ 600 → a ≥ 33.33, so the minimum whole number is 34. Answer: A
Multi-Step · Hard · Question 54
A factory’s daily cost is C = 500 + 8u, where u is units produced. Revenue is R = 20u. What is the minimum whole number of units needed for revenue to exceed cost by more than $1,000?
A) 126
B) 125
C) 124
D) 130
Show full solution
R − C > 1000 → 20u − (500 + 8u) > 1000 → 12u − 500 > 1000 → 12u > 1500 → u > 125, so the minimum whole number is 126. Answer: A
Multi-Step · Hard · Question 55
A moving company estimates a job will take between 4 and 7 hours, inclusive. The rate is $85 per hour plus a $60 truck fee. What is the range of possible total charges T?
A) $400 ≤ T ≤ $655
B) $340 ≤ T ≤ $595
C) $400 ≤ T ≤ $595
D) $340 ≤ T ≤ $655
Show full solution
T = 60 + 85h where 4 ≤ h ≤ 7. Minimum: 60 + 340 = 400. Maximum: 60 + 595 = 655. Answer: A
Common Mistakes on SAT Inequality Word Problems
| Mistake | Why It Hurts | What to Do Instead |
|---|---|---|
| Swapping fixed and variable terms | Puts the wrong quantity next to the variable | Identify which cost never changes before writing the inequality |
| Forgetting to round to a whole number | Counts of people, boxes, or tickets can’t be fractional | Round down for “at most” limits, round up for “at least” minimums |
| Answering the wrong quantity | The question may ask for a related value, not x itself | Re-read the final sentence of the problem after solving |
| Misreading “more than” as “at least” | Strict inequalities and inclusive ones give different boundary answers | “More than / fewer than” = strict; “at least / at most” = inclusive |
| Comparing two plans backward | Leads to reversing which option is cheaper or better | Write “Plan A < Plan B” only after confirming which one the question calls cheaper |
How Should You Study SAT Word Problems in 2 Weeks?
| Days | Focus | What to Practice |
|---|---|---|
| Days 1–3 | Budget & cost problems | Separate fixed vs. variable costs before solving |
| Days 4–6 | Capacity & threshold problems | Practice rounding correctly for whole-number answers |
| Days 7–9 | Comparing two plans | Set up “Plan A vs. Plan B” inequalities from scratch |
| Days 10–12 | Multi-step & mixed problems | Combine two constraints into one compound inequality |
| Days 13–14 | Timed mixed review | 20 mixed word problems under time pressure; review every missed translation |
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Frequently Asked Questions About SAT Inequality Word Problems
Why are word problems harder than direct-solving inequality questions?
Typically, the algebra is straightforward; the challenge lies in converting common language into the appropriate fixed term, variable term, and inequality sign before any solving can place.
How do I know whether to round up or down in a word problem?
For “at most” or “no more than” restrictions, round down because exceeding the precise figure would violate the restriction. For “at least” or minimal requirements, round up because not meeting the target will result in failure
What’s the fastest way to set up a “comparing two plans” problem?
After writing each plan as a separate cost expression, arrange the inequality sign between them according to which plan the question indicates is larger, faster, or less expensive.
Do I need to memorize keyword-to-symbol translations?
Indeed, “more than” and “fewer than” are stringent, while “at least” and “at most” contain the border. Every word issue takes less time when you are aware of these.
How many word problems should I practice before test day?
Before the test, most students find it beneficial to study through 40–55 mixed-word problems that include thresholds, capacity constraints, budgets, and plan comparisons.
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