Algebra 2 Quiz: Modeling With Equation Inequalityconstraints
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Modeling With Equation InequalityconstraintsQuestion 1 of 20

A rectangle is being designed for a poster. Let LL = length (in inches) and WW = width (in inches). The perimeter must be at most 40 inches, and the area must be at least 84 square inches. Also L>0L>0 and W>0W>0. Which system correctly represents these constraints?

{2L+2W≤40LW≥84L>0,W>0\begin{cases}2L+2W\le 40\\LW\ge 84\\L>0,\\ W>0\end{cases}
{2L+2W≥40LW≥84L>0,W>0\begin{cases}2L+2W\ge 40\\LW\ge 84\\L>0,\\ W>0\end{cases}
{L+W≤40LW≥84L>0,W>0\begin{cases}L+W\le 40\\LW\ge 84\\L>0,\\ W>0\end{cases}
{2L+2W≤40LW≤84L>0,W>0\begin{cases}2L+2W\le 40\\LW\le 84\\L>0,\\ W>0\end{cases}
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Algebra 2 Quiz: Modeling With Equation Inequalityconstraints

Practice Modeling With Equation Inequalityconstraints in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Modeling With Equation Inequalityconstraints, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A rectangle is being designed for a poster. Let LL = length (in inches) and WW = width (in inches). The perimeter must be at most 40 inches, and the area must be at least 84 square inches. Also L>0L>0 and W>0W>0. Which system correctly represents these constraints?

  1. {2L+2W≤40LW≥84L>0,W>0\begin{cases}2L+2W\le 40\\LW\ge 84\\L>0,\\ W>0\end{cases} (correct answer)
  2. {2L+2W≥40LW≥84L>0,W>0\begin{cases}2L+2W\ge 40\\LW\ge 84\\L>0,\\ W>0\end{cases}
  3. {L+W≤40LW≥84L>0,W>0\begin{cases}L+W\le 40\\LW\ge 84\\L>0,\\ W>0\end{cases}
  4. {2L+2W≤40LW≤84L>0,W>0\begin{cases}2L+2W\le 40\\LW\le 84\\L>0,\\ W>0\end{cases}
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. A constraint is a limitation or requirement: 'budget at most $500' becomes cost ≤ 500, 'need at least 10 units' becomes quantity ≥ 10. The system of ALL constraints defines the feasible region—the set of all solutions that work. A solution is viable if it lies in this feasible region (satisfies every single inequality/equation) AND makes real-world sense (no negative quantities, whole units when needed, etc.). Even one violation makes it nonviable! The constraints are perimeter 2L + 2W ≤ 40 (at most 40 inches), area LW ≥ 84 (at least 84 sq in), and L > 0, W > 0 (positive dimensions). Choice A correctly represents all constraints with the full perimeter formula and ≥ for area. A distractor like Choice C uses L + W ≤ 40, which is only half the perimeter—remember perimeter is 2(L + W), so include the 2! Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 2

A student council is assembling care kits using granola bars and juice boxes. Let xx = number of granola bars and yy = number of juice boxes.

Constraints:

  • Granola bars cost $1 each and juice boxes cost $2 each; budget is at most $50: x+2y≤50x+2y\le 50.
  • They need at least 30 total items: x+y≥30x+y\ge 30.
  • They must include at least 12 juice boxes: y≥12y\ge 12.
  • Items must be whole numbers and non-negative.

Which of the following is nonviable?

  1. (x,y)=(26,12)(x,y)=(26,12)
  2. (x,y)=(10,20)(x,y)=(10,20)
  3. (x,y)=(6,12)(x,y)=(6,12) (correct answer)
  4. (x,y)=(24,13)(x,y)=(24,13)
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! Let's check each option: For A (26,12): x + 2y = 26 + 2(12) = 26 + 24 = 50 ≤ 50 ✓, x + y = 26 + 12 = 38 ≥ 30 ✓, y = 12 ≥ 12 ✓. For B (10,20): x + 2y = 10 + 2(20) = 10 + 40 = 50 ≤ 50 ✓, x + y = 10 + 20 = 30 ≥ 30 ✓, y = 20 ≥ 12 ✓. For C (6,12): x + 2y = 6 + 2(12) = 6 + 24 = 30 ≤ 50 ✓, x + y = 6 + 12 = 18 < 30 ✗ (violates minimum total). For D (24,13): x + 2y = 24 + 2(13) = 24 + 26 = 50 ≤ 50 ✓, x + y = 24 + 13 = 37 ≥ 30 ✓, y = 13 ≥ 12 ✓. Choice C correctly identifies (6,12) as nonviable because it violates the constraint x + y ≥ 30, with only 18 total items instead of the required 30. The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. Thorough constraint checking is what separates good modeling from sloppy work!

Question 3

A smoothie shop is mixing two smoothies: fruit smoothie and protein smoothie. Let xx = number of fruit smoothies and yy = number of protein smoothies.

Constraints:

  • Each fruit smoothie uses 1 cup of fruit; each protein smoothie uses 2 cups of fruit; there are at most 18 cups of fruit: x+2y≤18x+2y\le 18.
  • Each fruit smoothie uses 0 scoops of protein powder; each protein smoothie uses 1 scoop; there are at most 7 scoops: y≤7y\le 7.
  • They want to make at least 10 smoothies total: x+y≥10x+y\ge 10.
  • Non-negativity: x≥0x\ge 0, y≥0y\ge 0.

Check if (x,y)=(6,7)(x,y)=(6,7) satisfies all constraints.

  1. Viable: it satisfies all constraints.
  2. Nonviable: it violates x+2y≤18x+2y\le 18. (correct answer)
  3. Nonviable: it violates y≤7y\le 7.
  4. Nonviable: it violates x+y≥10x+y\ge 10.
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! Let's check (x,y) = (6,7) against each constraint: (1) Fruit cups: x + 2y = 6 + 2(7) = 6 + 14 = 20. We need 20 ≤ 18, but 20 > 18 ✗ (violates fruit constraint). (2) Protein scoops: y = 7 ≤ 7 ✓. (3) Total smoothies: x + y = 6 + 7 = 13 ≥ 10 ✓. (4) Non-negativity: 6 ≥ 0 ✓ and 7 ≥ 0 ✓. Choice B correctly identifies that the point violates x + 2y ≤ 18, as 20 > 18. Even though all other constraints are satisfied, this single violation makes the solution nonviable. The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 4

A parking garage has two types of spaces: regular spaces that are 88 feet wide and compact spaces that are 77 feet wide. The garage has 180180 feet of width available for one row of parking spaces. City regulations require that at least 60%60\% of spaces must be regular spaces. The garage wants to maximize the total number of parking spaces in this row. If rr represents regular spaces and cc represents compact spaces, which constraint correctly represents the city regulation requirement?

  1. r+c≥0.6rr + c \geq 0.6r because total spaces must exceed 60%60\% of regular spaces
  2. r≥0.6(r+c)r \geq 0.6(r + c) because regular spaces must be at least 60%60\% of total spaces (correct answer)
  3. r≥0.6cr \geq 0.6c because regular spaces must exceed 60%60\% of compact spaces
  4. 0.6r+c≤r+c0.6r + c \leq r + c because the weighted total cannot exceed available spaces
Explanation: When translating percentage requirements into mathematical constraints, you need to identify what quantity represents the "whole" and what represents the "part." Here, the city requires at least 60% of all parking spaces to be regular spaces. The correct constraint is r≥0.6(r+c)r \geq 0.6(r + c) because regular spaces (rr) must be at least 60% of the total spaces (r+cr + c). This directly translates the requirement: "regular spaces ≥ 60% of total spaces." Let's examine why each wrong answer fails. Choice A states r+c≥0.6rr + c \geq 0.6r, which says total spaces must exceed 60% of regular spaces. This reverses the relationship entirely—it's comparing total spaces to a fraction of regular spaces rather than requiring regular spaces to be a fraction of the total. Choice C gives r≥0.6cr \geq 0.6c, meaning regular spaces must exceed 60% of compact spaces. This compares regular to compact spaces only, ignoring the total count that the regulation actually references. Choice D presents 0.6r+c≤r+c0.6r + c \leq r + c, which is always true (since 0.6r<r0.6r < r) and relates to the width constraint rather than the percentage requirement. When working with percentage constraints in optimization problems, always identify the base quantity (what the percentage is "of") and the quantity being constrained. Write it as: [constrained quantity] ≥ [percentage] × [base quantity]. This systematic approach prevents confusion between comparing parts to wholes versus parts to other parts.

Question 5

A electronics manufacturer produces smartphones and tablets. Due to component shortages, they can make at most 300300 devices per week. Smartphones require 22 hours of assembly time each, while tablets require 33 hours each. The company has 750750 hours of assembly time available weekly. Market demand requires producing at least twice as many smartphones as tablets. Which inequality represents the market demand constraint if ss is smartphones and tt is tablets?

  1. s+t≥2ts + t \geq 2t because total production must include twice as many smartphones
  2. 2s≥t2s \geq t because twice the smartphone production must exceed tablet production
  3. s≥2ts \geq 2t because smartphones must be at least double the tablet quantity (correct answer)
  4. t≤s2t \leq \frac{s}{2} because tablet production is limited to half the smartphone production
Explanation: When you encounter linear programming problems involving constraints, focus on translating the word problem into mathematical inequalities by identifying the key relationships between variables. The market demand constraint states that the company must produce "at least twice as many smartphones as tablets." This means smartphone production (ss) must be greater than or equal to two times tablet production (tt). Mathematically, this translates directly to s≥2ts \geq 2t. Let's examine why the other options miss the mark: Option A (s+t≥2ts + t \geq 2t) simplifies to s≥ts \geq t, which only requires smartphones to equal or exceed tablets, not be twice as many. This misinterprets the constraint by including total production rather than comparing the individual quantities. Option B (2s≥t2s \geq t) reverses the relationship entirely. This would mean twice the smartphones must exceed tablets, but the problem states smartphones must be at least twice the tablets, not that double the smartphones exceeds tablets. Option D (t≤s2t \leq \frac{s}{2}) is mathematically equivalent to the correct answer when rearranged, but it's written from the tablet perspective rather than directly expressing the smartphone constraint as stated in the problem. Study tip: When translating "at least twice as many of A as B," write it as A≥2BA \geq 2B. Practice converting word constraints into inequalities by identifying which variable should be isolated and whether the relationship requires "at least" (≥), "at most" (≤), or exact equality (=).

Question 6

A rectangle is being designed for a poster. Let LL = length (in inches) and WW = width (in inches).

Constraints:

  • Perimeter must be at most 40 inches.
  • Area must be at least 75 square inches.
  • Both dimensions must be positive: L>0, W>0L>0,\ W>0.

Which system represents these constraints?

  1. {2L+2W≥40LW≥75L>0,W>0\begin{cases}2L+2W\ge 40\\LW\ge 75\\L>0,\\ W>0\end{cases}
  2. {2L+2W≤40LW≥75L>0,W>0\begin{cases}2L+2W\le 40\\LW\ge 75\\L>0,\\ W>0\end{cases} (correct answer)
  3. {2L+2W≤40LW≤75L>0,W>0\begin{cases}2L+2W\le 40\\LW\le 75\\L>0,\\ W>0\end{cases}
  4. {L+W≤40LW≥75L>0,W>0\begin{cases}L+W\le 40\\LW\ge 75\\L>0,\\ W>0\end{cases}
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. A constraint is a limitation or requirement: 'budget at most $500' becomes cost ≤ 500, 'need at least 10 units' becomes quantity ≥ 10. The system of ALL constraints defines the feasible region—the set of all solutions that work. A solution is viable if it lies in this feasible region (satisfies every single inequality/equation) AND makes real-world sense (no negative quantities, whole units when needed, etc.). Even one violation makes it nonviable! The constraints are perimeter 2L + 2W ≤ 40 (at most 40), area LW ≥ 75 (at least 75), and positivity L > 0, W > 0. Choice B correctly represents all constraints with the complete system, using ≤ for perimeter and ≥ for area. Choice A incorrectly uses ≥ for perimeter, but 'at most' means ≤, which would require too large a perimeter. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 7

A school club is assembling care packages using granola bars and juice boxes. Let xx = number of granola bars and yy = number of juice boxes. Each granola bar costs $1.20 and each juice box costs $0.80. The club has at most $60 to spend, needs at least 50 total items, and cannot include more than 40 juice boxes. Quantities must be whole numbers and nonnegative. Which system of inequalities (with context constraints) represents this situation?

  1. {1.2x+0.8y≥60x+y≥50y≤40x≥0,y≥0\begin{cases}1.2x+0.8y\ge 60\\x+y\ge 50\\y\le 40\\x\ge 0,\\ y\ge 0\end{cases}
  2. {1.2x+0.8y≤60x+y≥50y≤40x≥0,y≥0x,y∈Z\begin{cases}1.2x+0.8y\le 60\\x+y\ge 50\\y\le 40\\x\ge 0,\\ y\ge 0\\x,y\in\mathbb{Z}\end{cases} (correct answer)
  3. {1.2x+0.8y≤60x+y≤50y≤40x≥0,y≥0x,y∈Z\begin{cases}1.2x+0.8y\le 60\\x+y\le 50\\y\le 40\\x\ge 0,\\ y\ge 0\\x,y\in\mathbb{Z}\end{cases}
  4. {2x+y≤60x+y≥50y≤40x≥0,y≥0x,y∈Z\begin{cases}2x+y\le 60\\x+y\ge 50\\y\le 40\\x\ge 0,\\ y\ge 0\\x,y\in\mathbb{Z}\end{cases}
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. A constraint is a limitation or requirement: 'budget at most $500' becomes cost ≤ 500, 'need at least 10 units' becomes quantity ≥ 10. The system of ALL constraints defines the feasible region—the set of all solutions that work. A solution is viable if it lies in this feasible region (satisfies every single inequality/equation) AND makes real-world sense (no negative quantities, whole units when needed, etc.). Even one violation makes it nonviable! The constraints here are cost 1.2x + 0.8y ≤ 60 (at most $60), total items x + y ≥ 50 (at least 50), juice boxes y ≤ 40 (no more than 40), and x ≥ 0, y ≥ 0 with x, y integers (nonnegative whole numbers). Choice B correctly represents all constraints with the proper inequalities for cost (≤), total (≥), and juice (≤), including integer and nonnegative conditions. A common distractor like Choice A flips the cost to ≥, which would mean spending at least $60, but the situation requires at most $60—always double-check the direction of inequalities! Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 8

A theater is selling two types of tickets: adult and student. Let aa = number of adult tickets and ss = number of student tickets.

Constraints:

  • The theater can seat at most 180 people: a+s≤180a+s\le 180.
  • At least 40 student tickets must be sold: s≥40s\ge 40.
  • Revenue goal: adult tickets are $15 and student tickets are $10; total revenue must be at least $2200.
  • Nonnegativity: a≥0, s≥0a\ge 0,\ s\ge 0.

Which of the following points is viable?

  1. $(a,s)=(120,50) (correct answer)
  2. $(a,s)=(100,30)
  3. $(a,s)=(160,40)
  4. $(a,s)=(150,35)
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! Checking each: (120,50) seats 170 ≤ 180 (✓), s=50 ≥ 40 (✓), revenue 15(120)+10(50)=1800+500=2300 ≥ 2200 (✓); (100,30) s=30 < 40 (✗); (160,40) seats 200 > 180 (✗); (150,35) seats 185 > 180 (✗) and s=35 < 40 (✗)—only (120,50) works. Choice A correctly determines viability with complete system checking for all points. Choice B gently, but (100,30) violates minimum students, though others fail differently—always check all. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 9

A student is choosing smoothies. Let xx = number of protein smoothies and yy = number of fruit smoothies.

Constraints:

  • Protein requirement: at least 30 g total. Protein smoothie has 12 g; fruit smoothie has 6 g.
  • Calorie limit: at most 500 calories total. Protein smoothie has 220 cal; fruit smoothie has 140 cal.
  • Nonnegativity: x≥0, y≥0x\ge 0,\ y\ge 0.

Determine which constraint the point (x,y)=(1,4)(x,y)=(1,4) violates (if any).

  1. It violates the protein requirement.
  2. It violates the calorie limit. (correct answer)
  3. It violates nonnegativity.
  4. It violates none of the constraints (it is feasible).
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (1,4), protein: 12(1) + 6(4) = 12 + 24 = 36 ≥ 30 (✓); calories: 220(1) + 140(4) = 220 + 560 = 780 ≤ 500? 780 > 500 (✗); nonnegativity: yes—so violates calories. Choice B correctly identifies the violated constraint with proper reasoning on the calorie check. Choice A gently, but it actually violates calories, as the substitution shows 780 > 500, so not protein. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 10

A community garden is planning crops. Let cc = acres of corn and ss = acres of soybeans.\n\nConstraints:\n- Total land available: at most 60 acres: c+s≤60c+s\le 60.\n- Water available: at most 100 acre-ft. Corn uses 2 acre-ft per acre; soybeans use 1.5 acre-ft per acre.\n- At least 10 acres of corn must be planted.\n- Nonnegativity: c≥0c\ge 0, s≥0s\ge 0.\n\nWhich system correctly represents all constraints?

  1. {c+s≤602c+1.5s≤100c≤10c≥0,s≥0\begin{cases}c+s\le 60\\2c+1.5s\le 100\\c\le 10\\c\ge 0,\\ s\ge 0\end{cases}
  2. {c+s≤602c+1.5s≥100c≥10c≥0,s≥0\begin{cases}c+s\le 60\\2c+1.5s\ge 100\\c\ge 10\\c\ge 0,\\ s\ge 0\end{cases}
  3. {c+s≥602c+1.5s≤100c≥10c≥0,s≥0\begin{cases}c+s\ge 60\\2c+1.5s\le 100\\c\ge 10\\c\ge 0,\\ s\ge 0\end{cases}
  4. {c+s≤602c+1.5s≤100c≥10c≥0,s≥0\begin{cases}c+s\le 60\\2c+1.5s\le 100\\c\ge 10\\c\ge 0,\\ s\ge 0\end{cases} (correct answer)
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. A constraint is a limitation or requirement: 'budget at most $500' becomes $cost \leq 500,′needatleast10units′becomes, 'need at least 10 units' becomes quantity \geq 10.ThesystemofALLconstraintsdefinesthefeasibleregion—thesetofallsolutionsthatwork.Asolutionisviableifitliesinthisfeasibleregion(satisfieseverysingleinequality/equation)ANDmakesreal−worldsense(nonegativequantities,wholeunitswhenneeded,etc.).Evenoneviolationmakesitnonviable!Theconstraintsareland. The system of ALL constraints defines the feasible region—the set of all solutions that work. A solution is viable if it lies in this feasible region (satisfies every single inequality/equation) AND makes real-world sense (no negative quantities, whole units when needed, etc.). Even one violation makes it nonviable! The constraints are land c + s \leq 60,water, water 2c + 1.5s \leq 100,minimumcorn, minimum corn c \geq 10,andnonnegativity, and nonnegativity c \geq 0,, s \geq 0,matchingthegivendetailsexactly.ChoiceAcorrectlyrepresentsallconstraintswiththecompletesystem,using, matching the given details exactly. Choice A correctly represents all constraints with the complete system, using \leqforbothlimitedresources.ChoiceBincorrectlyflipsthelandtofor both limited resources. Choice B incorrectly flips the land to\geq,but′atmost′means, but 'at most' means \leq,notrequiringmorelandthanavailable.Constraintidentificationfromcontext:(1)listeverylimitationmentioned(′budget, not requiring more land than available. Constraint identification from context: (1) list every limitation mentioned ('budget X,' 'time \leqY hours,' 'need\geqZunits′),(2)translateusingkeyphrases:′atmost′Z units'), (2) translate using key phrases: 'at most'\to \leq,′atleast′, 'at least' \to \geq,′exactly′, 'exactly' \to =,′morethan′, 'more than' \to >,′lessthan′, 'less than' \to <,(3)don′tforgetimplicitconstraintslike, (3) don't forget implicit constraints like x \geq 0,, y \geq 0(can′tbenegative)or(can't be negative) orx,, yintegers(ifdiscrete),(4)writethecompletesystem.Missingevenoneconstraintcanmakeyouacceptinfeasiblesolutions!Theviabilitychecklist:Makeatablewithonerowperconstraint.Foreachconstraint,substituteyourpointandmarkwhetherit′ssatisfied( integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (\checkmark)orviolated() or violated (\times).IfallcheckspassANDcontextisreasonable,markVIABLE.Ifevenone). If all checks pass AND context is reasonable, mark VIABLE. If even one \times$ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 11

A student has a monthly streaming budget and time limit. Let xx = hours of Service A watched and yy = hours of Service B watched. Service A costs $2 per hour and Service B costs $1 per hour. The student can spend at most $30 and watch at most 25 hours total. The student also wants at least 8 hours of Service A. Which system represents the constraints?

  1. {2x+y≤30x+y≤25x≥8x≥0,y≥0\begin{cases}2x+y\le 30\\x+y\le 25\\x\ge 8\\x\ge 0,\\ y\ge 0\end{cases} (correct answer)
  2. {2x+y≥30x+y≤25x≥8x≥0,y≥0\begin{cases}2x+y\ge 30\\x+y\le 25\\x\ge 8\\x\ge 0,\\ y\ge 0\end{cases}
  3. {2x+y≤30x+y≥25x≥8x≥0,y≥0\begin{cases}2x+y\le 30\\x+y\ge 25\\x\ge 8\\x\ge 0,\\ y\ge 0\end{cases}
  4. {2x+y≤30x+y≤25x≤8x≥0,y≥0\begin{cases}2x+y\le 30\\x+y\le 25\\x\le 8\\x\ge 0,\\ y\ge 0\end{cases}
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. A constraint is a limitation or requirement: 'budget at most $500' becomes cost ≤ 500, 'need at least 10 units' becomes quantity ≥ 10. The system of ALL constraints defines the feasible region—the set of all solutions that work. A solution is viable if it lies in this feasible region (satisfies every single inequality/equation) AND makes real-world sense (no negative quantities, whole units when needed, etc.). Even one violation makes it nonviable! The constraints are cost 2x + y ≤ 30 (at most $30), total x + y ≤ 25 (at most 25 hours), x ≥ 8 (at least 8 of A), and x ≥ 0, y ≥ 0 (nonnegative). Choice A correctly represents all constraints with ≤ for cost and total, and ≥ for minimum A. A distractor like Choice C flips total to ≥25, but it's at most 25—match 'at most' to ≤ and 'at least' to ≥ thoughtfully! Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 12

A delivery company is packing boxes with two item types. Let xx = number of Type A items and yy = number of Type B items. A box can hold at most 30 items total: x+y≤30x+y\le 30. Type A items weigh 2 lb each and Type B items weigh 1 lb each, and the box can weigh at most 48 lb: 2x+y≤482x+y\le 48. Also, at least 12 Type B items must be included: y≥12y\ge 12. Which point is viable?

  1. (25,12)(25,12)
  2. (10,25)(10,25)
  3. (18,12)(18,12) (correct answer)
  4. (20,10)(20,10)
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (18,12): total 18 + 12 = 30 ≤ 30 (ok), weight 2*18 + 12 = 36 + 12 = 48 ≤ 48 (ok), y=12 ≥ 12 (ok); others like (20,10) has y=10 < 12 (violated), (25,12) 37 > 30 (violated), (10,25) 35 > 30 (violated)—only (18,12) works. Choice B correctly determines viability with proper checking of all points against every constraint. A distractor like Choice A might accept (20,10) by missing y ≥ 12, but it violates—check minimum requirements carefully! Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 13

A community garden is deciding how many acres of tomatoes and peppers to plant. Let xx be acres of tomatoes and yy be acres of peppers.

Constraints:

  • Total land available: at most 12 acres: x+y≤12x+y\le 12.
  • Water limit: tomatoes need 3 units/acre and peppers need 2 units/acre; at most 30 units: 3x+2y≤303x+2y\le 30.
  • The garden must plant at least 2 acres of peppers: y≥2y\ge 2.
  • Non-negativity: x≥0x\ge 0, y≥0y\ge 0.

Which point is viable?

  1. (8,1)(8,1)
  2. (6,4)(6,4) (correct answer)
  3. (10,3)(10,3)
  4. (11,2)(11,2)
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (6,4), land: 6+4=10≤12 (✓), water: 3(6)+2(4)=18+8=26≤30 (✓), peppers: 4≥2 (✓), x≥0 y≥0 (✓)—all pass; others like (10,3) fail land 13>12. Choice C correctly determines viability with proper reasoning on the only point satisfying all. A distractor like choice A might overlook land 10+3=13>12, but gently verify each substitution to catch that. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 14

A rectangular dog run must satisfy two design constraints. Let LL be the length (ft) and WW be the width (ft).

Constraints:

  • Fencing available: perimeter at most 60 ft: 2L+2W≤602L+2W\le 60.
  • The area must be at least 200 square feet: LW≥200LW\ge 200.
  • Non-negativity: L≥0L\ge 0, W≥0W\ge 0.

Is (L,W)=(20,10)(L,W)=(20,10) viable?

  1. Viable: it satisfies perimeter and area constraints. (correct answer)
  2. Nonviable: it violates the area constraint.
  3. Nonviable: it violates the perimeter constraint.
  4. Nonviable: it violates non-negativity.
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (20,10), perimeter: 2(20)+2(10)=40+20=60≤60 (✓), area: 20*10=200≥200 (✓), nonneg (✓)—all pass, so viable. Choice A correctly determines viability with proper reasoning, including equality allowed in both constraints. A distractor like choice B might miscalculate perimeter as over, but gently substitute to see 60≤60 holds. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 15

A student is choosing how many hours to spend tutoring in math and science this week. Let xx be hours of math tutoring and yy be hours of science tutoring.

Constraints:

  • Total tutoring time is at most 12 hours: x+y≤12x+y\le 12.
  • The student wants at least twice as much math as science: x≥2yx\ge 2y.
  • At least 3 hours of science: y≥3y\ge 3.
  • Non-negativity: x≥0x\ge 0, y≥0y\ge 0.

Determine which solutions are viable: (6,3)(6,3), (5,3)(5,3), (8,4)(8,4).

  1. Only (8,4)(8,4) is viable.
  2. All three points are viable.
  3. (6,3)(6,3) and (8,4)(8,4) are viable. (correct answer)
  4. Only (6,3)(6,3) is viable.
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (6,3): 6+3=9≤12, 6≥2*3=6, 3≥3 (all ✓); for (8,4): 8+4=12≤12, 8≥8, 4≥3 (all ✓); but (5,3): 5<6 violates x≥2y. Choice C correctly determines viability with proper reasoning for both points. A distractor like choice A might miss checking (8,4) fully, but gently substitute into each constraint for all points to confirm. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 16

An event planner is ordering chairs and tables for a fundraiser. Let xx be the number of chairs and yy be the number of tables.

Constraints:

  • Each chair costs $8 and each table costs $25; budget at most $400: 8x+25y≤4008x+25y\le 400.
  • Seating requirement: each table requires 4 chairs, so there must be at least 4 chairs per table: x≥4yx\ge 4y.
  • At least 6 tables are needed: y≥6y\ge 6.
  • Chairs and tables must be whole numbers and nonnegative.

Which constraint does (x,y)=(20,7)(x,y)=(20,7) violate?

  1. Integer/nonnegativity constraints
  2. Budget constraint 8x+25y≤4008x+25y\le 400
  3. Seating requirement x≥4yx\ge 4y (correct answer)
  4. Minimum tables y≥6y\ge 6
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (20,7), budget: 8(20)+25(7)=160+175=335≤400 (✓), seating: 20≥4*7=28? 20<28 (✗), tables: 7≥6 (✓), integers nonneg (✓)—violates seating. Choice B correctly identifies the violated constraint with complete checking. A distractor like choice A might ignore seating and only check budget, but gently remember to test all constraints systematically. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 17

A student is mixing two types of paint to make custom posters. Let xx be liters of Paint A and yy be liters of Paint B.

Constraints:

  • Total paint needed is at least 5 liters: x+y≥5x+y\ge 5.
  • Paint A is limited to at most 3 liters: x≤3x\le 3.
  • The mixture must have at least as much Paint B as Paint A: y≥xy\ge x.
  • Non-negativity: x≥0x\ge 0, y≥0y\ge 0.

Which point is in the feasible region (satisfies all constraints)?

  1. (3,1)(3,1)
  2. (2,2)(2,2)
  3. (3,3)(3,3) (correct answer)
  4. (4,4)(4,4)
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (3,3), total: 3+3=6≥5 (✓), x: 3≤3 (✓), y≥x: 3≥3 (✓), nonneg (✓)—all pass; others like (3,1) have 4<5. Choice C correctly determines viability with proper reasoning on the point satisfying all. A distractor like choice A might miss total 3+1=4<5, but gently check each inequality after substituting. Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 18

A bakery is deciding how many loaves of bread and batches of muffins to make. Let xx = loaves of bread and yy = batches of muffins. Each loaf uses 0.5 kg of flour and each muffin batch uses 0.8 kg of flour. The bakery has at most 20 kg of flour. The oven time limit is 30 hours; each loaf takes 1 hour and each muffin batch takes 2 hours. They must make at least 10 loaves of bread. Which constraint does the point (x,y)=(10,12)(x,y)=(10,12) violate, if any? (Assume nonnegative integers.)

  1. It violates the oven-time constraint. (correct answer)
  2. It violates the minimum bread requirement.
  3. It violates the flour constraint.
  4. It does not violate any constraint (it is viable).
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! For (10,12): flour 0.510 + 0.812 = 5 + 9.6 = 14.6 ≤ 20 (satisfied), oven 110 + 212 = 10 + 24 = 34 > 30 (violated), bread 10 ≥ 10 (satisfied), nonnegative integers (yes)—violates oven time. Choice B correctly identifies the violated constraint with complete substitution. A distractor like Choice D might skip checking oven as 34 ≤ 30, but it's >—ensure you evaluate all inequalities fully! Constraint identification from context: (1) list every limitation mentioned ('budget $X,' 'time ≤ Y hours,' 'need ≥ Z units'), (2) translate using key phrases: 'at most' → ≤, 'at least' → ≥, 'exactly' → =, 'more than' → >, 'less than' → <, (3) don't forget implicit constraints like x ≥ 0, y ≥ 0 (can't be negative) or x, y integers (if discrete), (4) write the complete system. Missing even one constraint can make you accept infeasible solutions! The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 19

A food truck plans to make tacos (x) and salads (y). Each taco takes 2 minutes to assemble and each salad takes 5 minutes. The truck has at most 180 minutes of prep time. For menu balance, they must make at least 10 salads. Also x≥0x\ge 0 and y≥0y\ge 0.

Is the plan (x,y)=(60,10)(x,y)=(60,10) viable? (A plan is viable if it satisfies all constraints.)

  1. Viable: 2(60)+5(10)=170≤1802(60)+5(10)=170\le 180 and 10≥1010\ge 10. (correct answer)
  2. Nonviable: it violates the prep-time constraint because 2(60)+5(10)=190>1802(60)+5(10)=190>180.
  3. Nonviable: it violates the salad minimum because 10<1010<10.
  4. Viable: any nonnegative (x,y)(x,y) is allowed since time is not a constraint.
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! Let's check (60, 10) against all constraints: (1) Time constraint: 2x + 5y ≤ 180 → 2(60) + 5(10) = 120 + 50 = 170 ≤ 180 ✓; (2) Salad minimum: y ≥ 10 → 10 ≥ 10 ✓; (3) Non-negativity: x ≥ 0 → 60 ≥ 0 ✓ and y ≥ 0 → 10 ≥ 0 ✓. All constraints are satisfied! Choice A correctly determines viability by checking the time constraint (170 ≤ 180) and salad minimum (10 ≥ 10), confirming the plan is viable. Choice B incorrectly calculates 2(60) + 5(10) = 190, which is wrong (it's actually 170); Choice C incorrectly claims 10 < 10, when 10 = 10 satisfies y ≥ 10; Choice D incorrectly suggests time isn't a constraint when it clearly is. The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!

Question 20

A community garden is planting tomatoes (x) and peppers (y). The garden has at most 30 plots total, so x+y≤30x+y\le 30. Tomatoes require 3 units of compost per plot and peppers require 1 unit per plot; at most 60 compost units are available, so 3x+y≤603x+y\le 60. Also, at least 8 plots must be peppers: y≥8y\ge 8. Non-negativity applies.

Which constraint does the plan (x,y)=(18,10)(x,y)=(18,10) violate, if any?

  1. It violates x+y≤30x+y\le 30.
  2. It violates 3x+y≤603x+y\le 60. (correct answer)
  3. It violates y≥8y\ge 8.
  4. It violates no constraints; the plan is viable.
Explanation: This question tests your ability to translate real-world limitations into mathematical constraints (equations and inequalities) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in context. Checking viability is systematic: (1) write out each constraint, (2) substitute the proposed solution into each one, (3) verify each is satisfied (inequality holds, equation balances), (4) check context reasonableness (non-negative? integers if needed?). If everything passes, it's viable. If anything fails, it's nonviable—and you should identify WHICH constraint was violated. Complete checking means checking ALL constraints, not just some! Let's check (18, 10) against each constraint: (1) Total plots: x + y ≤ 30 → 18 + 10 = 28 ≤ 30 ✓; (2) Compost units: 3x + y ≤ 60 → 3(18) + 10 = 54 + 10 = 64 > 60 ✗ VIOLATED!; (3) Pepper minimum: y ≥ 8 → 10 ≥ 8 ✓; (4) Non-negativity: 18 ≥ 0 ✓ and 10 ≥ 0 ✓. The plan violates the compost constraint! Choice B correctly identifies that the plan violates 3x + y ≤ 60 because 3(18) + 10 = 64 > 60, making the plan nonviable. Choice A incorrectly claims it violates x + y ≤ 30, but 28 ≤ 30 is satisfied; Choice C incorrectly claims it violates y ≥ 8, but 10 ≥ 8 is satisfied; Choice D incorrectly claims the plan is viable when it clearly violates the compost constraint. The viability checklist: Make a table with one row per constraint. For each constraint, substitute your point and mark whether it's satisfied (✓) or violated (✗). If all checks pass AND context is reasonable, mark VIABLE. If even one ✗ appears OR context is violated (like negative values or fractional items), mark NONVIABLE and note which constraint failed. This organized approach prevents missing checks and makes your reasoning clear. Thorough constraint checking is what separates good modeling from sloppy work!