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!

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 $ \le 500$, 'need at least 10 units' becomes quantity $ \ge 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 $24x + 18y \le 180$ (no more than $180), serving $8x + 6y \ge 60$ (at least 60 people), and $x \ge 0$, $y \ge 0$ with integers (nonnegative whole trays). Choice A correctly represents all constraints with $ \le $ for cost and $ \ge $ for serving, including nonnegativity and integers. A distractor like Choice B flips cost to $ \ge 180$, which means spending at least $180, but the limit is at most—remember to match 'no more than' to $ \le $! Constraint identification from context: (1) list every limitation mentioned ('budget $X,$ 'time $ \le $ Y hours,' 'need $ \ge $ Z units'), (2) translate using key phrases: 'at most' $ \to \le $, 'at least' $ \to \ge $, 'exactly' $ \to =$, 'more than' $ \to >$, 'less than' $ \to <$, (3) don't forget implicit constraints like $x \ge 0$, $y \ge 0$ (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!

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 40x + 90y ≤ 900 (at most $900), total ads x + y ≥ 8 (at least 8), radio limit y ≤ 6, and nonnegativity x ≥ 0, y ≥ 0. Choice A correctly represents all constraints with the complete system, using the right directions for each. Choice D incorrectly uses y ≥ 6 for radio, but 'at most' is ≤, not requiring more. 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!

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!

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 12x + 5y ≤ 120 (at most $120), total items x + y ≥ 15 (at least 15), and nonnegativity x ≥ 0, y ≥ 0, forming the complete system. Choice B correctly represents all constraints with the complete system, including the correct direction for the budget inequality. Choice A incorrectly uses ≥ for the cost, but 'at most' translates to ≤, which would allow overspending. 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!

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!

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!

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 $cost \leq 500$, 'need at least 10 units' becomes $quantity \geq 10$. 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 land $c + s \leq 60$, water $2c + 1.5s \leq 100$, minimum corn $c \geq 10$, and nonnegativity $c \geq 0$, $s \geq 0$, matching the given details exactly. Choice A correctly represents all constraints with the complete system, using $\leq$ for both limited resources. Choice B incorrectly flips the land to $\geq$, but 'at most' means $\leq$, not requiring more land than available. Constraint identification from context: (1) list every limitation mentioned ('budget $X$,' 'time $\leq$ Y hours,' 'need $\geq$ Z units'), (2) translate using key phrases: 'at most' $\to \leq$, 'at least' $\to \geq$, 'exactly' $\to =$, 'more than' $\to >$, 'less than' $\to <$, (3) don't forget implicit constraints like $x \geq 0$, $y \geq 0$ (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 ($\checkmark$) or violated ($\times$). 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!

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: $(2,4)$ $2(2)+4=8 \le 12$ (✓), $2+2(4)=10 \le 14$ (✓); $(4,2)$ $2(4)+2=10 \le 12$ (✓), $4+2(2)=8 \le 14$ (✓); $(5,4)$ $2(5)+4=14>12$ (✗); $(0,7)$ $0+7=7 \le 12$ (✓), $0+2(7)=14 \le 14$ (✓)—only $(5,4)$ fails. Choice C correctly identifies the nonviable point with complete checking showing it violates $2x + y \le 12$. Choice A gently, but $(2,4)$ satisfies both inequalities, so it's viable—check substitutions 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 $x \ge 0$, $y \ge 0$ (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!

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!