This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. A solution is viable (feasible) if it satisfies EVERY SINGLE constraint AND makes sense in the real-world context (like no negative quantities, whole items when you can't buy half an item, etc.). If even one constraint is violated, or if the solution is unrealistic, it's nonviable. Think of constraints like security checkpoints—you need to pass through all of them! To check if (12, 6) is viable, we substitute into each constraint: Space constraint: 12 + 6 = 18 ≤ 30 ✓ (true). Minimum tomatoes: 12 ≥ 10 ✓ (true). Watering time: 2(12) + 3(6) = 24 + 18 = 42 ≤ 75 ✓ (true). Also checking context: 12 and 6 are both nonnegative whole numbers ✓. Conclusion: viable because all constraints are satisfied. Choice A is correct because it properly checks all constraints showing (12, 6) is viable and correctly identifies that all inequalities are satisfied. Choice B says the point is nonviable but doesn't check correctly: it claims 12 + 6 > 30, but 12 + 6 = 18, which is less than 30. To determine viability, you MUST check every single constraint accurately—missing even one or calculating incorrectly can lead to rejecting a perfectly good solution! The viability-checking procedure: Make a checklist of every constraint. For each one, substitute the point and check if it's satisfied. Write 'Yes' or 'No' next to each constraint. If even one 'No' appears, the solution is nonviable—identify which constraint(s) failed. Also do a reality check: negative quantities? fractional items when must be whole? These context violations also make solutions nonviable!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. When setting up constraints from context, look for key phrases: 'at most' or 'no more than' → ≤, 'at least' or 'minimum' → ≥, 'exactly' or 'must be' → =, 'less than' → <, 'more than' → >. Also, don't forget implicit constraints like x ≥ 0 and y ≥ 0 (can't have negative quantities) or x, y must be integers (if counting discrete items). Let's identify all the constraints from the snack-buying scenario: the budget 'at most $60' gives us 6x + 4y ≤ 60, the requirement 'at least 12 total boxes' gives us x + y ≥ 12, and non-negativity gives us x ≥ 0, y ≥ 0. Putting this together as a system: {6x + 4y ≤ 60, x + y ≥ 12, x ≥ 0, y ≥ 0}. This system captures all the limitations and requirements of the situation. Choice C is correct because it includes all constraints with correct inequality directions. Choice B is tempting but has the total boxes inequality backwards: 'at least 12' means ≥ 12, not ≤ 12, which would incorrectly limit to no more than 12 boxes instead of requiring a minimum. The constraint-writing recipe: (1) List EVERY limitation mentioned in the problem (budget, time, capacity, minimums, etc.), (2) For each, identify the inequality symbol from key words ('at most' → ≤, 'at least' → ≥, etc.), (3) Write the mathematical inequality using the costs, rates, or quantities from context, (4) Don't forget implicit constraints like x ≥ 0, y ≥ 0 (non-negativity) or x, y integers (if discrete). Make sure your system captures EVERY constraint!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. A solution is viable (feasible) if it satisfies EVERY SINGLE constraint AND makes sense in the real-world context (like no negative quantities, whole items when you can't buy half an item, etc.). If even one constraint is violated, or if the solution is unrealistic, it's nonviable. Think of constraints like security checkpoints—you need to pass through all of them! Let's check each point against all constraints: For (150, 120): Total tickets: 150 + 120 = 270 ≤ 300 ✓, Floor minimum: 120 ≥ 120 ✓, Non-negative: both ≥ 0 ✓ → viable. For (100, 200): Total tickets: 100 + 200 = 300 ≤ 300 ✓, Floor minimum: 200 ≥ 120 ✓, Non-negative: both ≥ 0 ✓ → viable. For (210, 110): Total tickets: 210 + 110 = 320 ≤ 300 ✗ (320 > 300), Floor minimum: 110 ≥ 120 ✗ (110 < 120) → nonviable (violates both!). For (0, 120): Total tickets: 0 + 120 = 120 ≤ 300 ✓, Floor minimum: 120 ≥ 120 ✓, Non-negative: both ≥ 0 ✓ → viable. Notice how (210, 110) fails because it violates BOTH constraints—too many total tickets AND not enough floor tickets! Choice C is correct because it correctly identifies the point (210, 110) as nonviable due to violating the total tickets constraint (320 > 300) and the floor tickets minimum (110 < 120). Choice A would be viable but (150, 120) actually satisfies all constraints: 150 + 120 = 270 ≤ 300 and 120 ≥ 120. Don't be fooled—a point needs to violate at least one constraint to be nonviable! The viability-checking procedure: Make a checklist of every constraint. For each one, substitute the point and check if it's satisfied. Write 'Yes' or 'No' next to each constraint. If even one 'No' appears, the solution is nonviable—identify which constraint(s) failed. When checking viability, substitute carefully: if the point is (210, 110), that means x = 210 and y = 110. Substitute those values into EVERY inequality and equation. It's tedious but necessary—one missed check could mean accepting an infeasible solution!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. A solution is viable (feasible) if it satisfies EVERY SINGLE constraint AND makes sense in the real-world context (like no negative quantities, whole items when you can't buy half an item, etc.). If even one constraint is violated, or if the solution is unrealistic, it's nonviable. Think of constraints like security checkpoints—you need to pass through all of them! To check if (3.5, 6) is viable, we substitute into each constraint: for 4x + y ≤ 25, 4(3.5) + 6 = 14 + 6 = 20 ≤ 25 (true); for x + y ≥ 8, 3.5 + 6 = 9.5 ≥ 8 (true); for x ≥ 0, y ≥ 0 (true). But checking context: must be whole numbers, and 3.5 is fractional, which is impossible for items. Conclusion: nonviable because violates whole-number requirement. Choice C is correct because it correctly identifies that it violates the whole-number requirement even though it satisfies the inequalities. Choice B says it's nonviable but incorrectly checks the budget: 20 ≤ 25 is satisfied, not violated. Remember, mathematical satisfaction isn't enough—context matters! The viability-checking procedure: Make a checklist of every constraint. For each one, substitute the point and check if it's satisfied. Write 'Yes' or 'No' next to each constraint. If even one 'No' appears, the solution is nonviable—identify which constraint(s) failed. Also do a reality check: negative quantities? fractional items when must be whole? These context violations also make solutions nonviable! Remember: viable means 'it could actually happen in real life.' Mathematically satisfying inequalities is necessary, but not sufficient. Ask yourself: In this context, can quantities be negative? Can they be fractions? Are there other real-world restrictions? A solution can satisfy all the math but still be nonviable if it violates reality!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. When setting up constraints from context, look for key phrases: 'at most' or 'no more than' → ≤, 'at least' or 'minimum' → ≥, 'exactly' or 'must be' → =, 'less than' → <, 'more than' → >. Also, don't forget implicit constraints like x ≥ 0 and y ≥ 0 (can't have negative quantities) or x, y must be integers (if counting discrete items). Let's identify all the constraints from the fundraiser situation: the budget 'at most $60' gives us 3x + 2y ≤ 60, the requirement 'at least 25 total items' gives us x + y ≥ 25, and the implicit non-negativity gives us x ≥ 0 and y ≥ 0. Putting this together as a system: {3x + 2y ≤ 60, x + y ≥ 25, x ≥ 0, y ≥ 0}. This system captures all the limitations and requirements of the situation. Choice C is correct because it includes all constraints with correct inequality directions. Choice A has the inequality direction backwards: 'at least 25' means ≥, not ≤. This is a super common error! When total items 'at least 25,' we need x + y ≥ 25, meaning we can't fall below 25. The 'at least' puts a floor, not a ceiling. The constraint-writing recipe: (1) List EVERY limitation mentioned in the problem (budget, time, capacity, minimums, etc.), (2) For each, identify the inequality symbol from key words ('at most' → ≤, 'at least' → ≥, etc.), (3) Write the mathematical inequality using the costs, rates, or quantities from context, (4) Don't forget implicit constraints like x ≥ 0, y ≥ 0 (non-negativity) or x, y integers (if discrete). Make sure your system captures EVERY constraint!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. Checking viability is systematic: (1) Substitute the proposed solution into each constraint, (2) Verify each inequality or equation is satisfied, (3) Check context realism (non-negative? whole numbers if needed?). If everything checks out, it's viable. If anything fails, it's nonviable—and you should explain which constraint was violated. To check if (30, 40) is viable, we substitute into each constraint: Checking 3x + y ≤ 120: 3(30) + 40 = 90 + 40 = 130 ≤ 120 ✗ (false, since 130 > 120). Checking x + y ≥ 20: 30 + 40 = 70 ≥ 20 ✓ (true). Checking y ≤ 50: 40 ≤ 50 ✓ (true). Checking x ≥ 0: 30 ≥ 0 ✓ and y ≥ 0: 40 ≥ 0 ✓. Conclusion: nonviable because it violates the time constraint 3x + y ≤ 120. Choice A is correct because it properly checks all constraints showing that only 3x + y ≤ 120 is violated (130 > 120). Choice D says the point is viable but doesn't check all constraints: it misses that 3(30) + 40 = 130 exceeds the 120-minute limit. To determine viability, you MUST check every single constraint—missing even one can lead to accepting an impossible solution! When checking viability, substitute carefully: if the point is (30, 40), that means x = 30 and y = 40. Substitute those values into EVERY inequality and equation. For example, if one constraint is 3x + y ≤ 120, check: 3(30) + 40 = 90 + 40 = 130 ≤ 120? No! Do this for every single constraint. It's tedious but necessary—one missed check could mean accepting an infeasible solution!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. When setting up constraints from context, look for key phrases: 'at most' or 'no more than' → ≤, 'at least' or 'minimum' → ≥, 'exactly' or 'must be' → =, 'less than' → <, 'more than' → >. Also, don't forget implicit constraints like x ≥ 0 and y ≥ 0 (can't have negative quantities) or x, y must be integers (if counting discrete items). Let's identify all the constraints from 'notebooks cost $3 each, pens cost $2 each, at most $60 to spend, need at least 10 total items': The budget constraint 'at most $60' gives us 3x + 2y ≤ 60. The minimum items constraint 'at least 10 total items' gives us x + y ≥ 10. The non-negativity constraints (can't buy negative items) give us x ≥ 0 and y ≥ 0. Putting this together as a system: {3x + 2y ≤ 60, x + y ≥ 10, x ≥ 0, y ≥ 0}. This system captures all the limitations and requirements of the situation. Choice B is correct because it includes all constraints with correct inequality directions: the budget constraint uses ≤ (at most), the minimum items constraint uses ≥ (at least), and non-negativity constraints are included. Choice A has the inequality direction backwards: 'at most $60' means ≤ (can equal or be less), not ≥. This is a super common error! When total cost 'at most $60,' we need cost ≤ 60, meaning we can't exceed 60. The 'at most' puts a ceiling, not a floor. The constraint-writing recipe: (1) List EVERY limitation mentioned in the problem (budget, time, capacity, minimums, etc.), (2) For each, identify the inequality symbol from key words ('at most' → ≤, 'at least' → ≥, etc.), (3) Write the mathematical inequality using the costs, rates, or quantities from context, (4) Don't forget implicit constraints like x ≥ 0, y ≥ 0 (non-negativity) or x, y integers (if discrete). Make sure your system captures EVERY constraint!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. A solution is viable (feasible) if it satisfies EVERY SINGLE constraint AND makes sense in the real-world context (like no negative quantities, whole items when you can't buy half an item, etc.). If even one constraint is violated, or if the solution is unrealistic, it's nonviable. Think of constraints like security checkpoints—you need to pass through all of them! To check if (9, 9) is viable, we substitute into each constraint: Checking 2x + y ≤ 30: 2(9) + 9 = 18 + 9 = 27 ≤ 30 ✓ (true). Checking x + y ≥ 18: 9 + 9 = 18 ≥ 18 ✓ (true). Checking x ≤ 10: 9 ≤ 10 ✓ (true). Checking x ≥ 0: 9 ≥ 0 ✓ and y ≥ 0: 9 ≥ 0 ✓ (true). Also checking context: both 9 and 9 are nonnegative whole numbers ✓. Conclusion: viable because all constraints are satisfied. Choice A is correct because it properly checks all constraints showing (9, 9) is viable—all inequalities are satisfied and the values make sense in context. Choice B incorrectly calculates: it says 2(9) + 9 = 27, which is correct, but then claims this violates the constraint when actually 27 ≤ 30 is true! When checking inequalities, be careful with the direction: ≤ means 'less than or equal to,' so 27 ≤ 30 is satisfied. The viability-checking procedure: Make a checklist of every constraint. For each one, substitute the point and check if it's satisfied. Write 'Yes' or 'No' next to each constraint. If even one 'No' appears, the solution is nonviable—identify which constraint(s) failed. Also do a reality check: negative quantities? fractional items when must be whole? These context violations also make solutions nonviable!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. When setting up constraints from context, look for key phrases: 'at most' or 'no more than' → ≤, 'at least' or 'minimum' → ≥, 'exactly' or 'must be' → =, 'less than' → <, 'more than' → >. Also, don't forget implicit constraints like x ≥ 0 and y ≥ 0 (can't have negative quantities) or x, y must be integers (if counting discrete items). Let's identify all the constraints from the farming situation: water 'at most 50 units' gives us 2x + y ≤ 50, total acres 'at most 30' gives us x + y ≤ 30, minimum corn 'at least 8 acres' gives us x ≥ 8, and non-negativity x ≥ 0, y ≥ 0. Putting this together as a system: {2x + y ≤ 50, x + y ≤ 30, x ≥ 8, x ≥ 0, y ≥ 0}. This system captures all the limitations and requirements of the situation. Choice A is correct because it includes all constraints with correct inequality directions. Choice C has the inequality direction backwards: 'at most 30 acres' means ≤, not ≥. This is a super common error! When total acres 'at most 30,' we need x + y ≤ 30, meaning we can't exceed 30. The 'at most' puts a ceiling, not a floor. The constraint-writing recipe: (1) List EVERY limitation mentioned in the problem (budget, time, capacity, minimums, etc.), (2) For each, identify the inequality symbol from key words ('at most' → ≤, 'at least' → ≥, etc.), (3) Write the mathematical inequality using the costs, rates, or quantities from context, (4) Don't forget implicit constraints like x ≥ 0, y ≥ 0 (non-negativity) or x, y integers (if discrete). Make sure your system captures EVERY constraint!

This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. Checking viability is systematic: (1) Substitute the proposed solution into each constraint, (2) Verify each inequality or equation is satisfied, (3) Check context realism (non-negative? whole numbers if needed?). If everything checks out, it's viable. If anything fails, it's nonviable—and you should explain which constraint was violated. To check if (6, 4) is viable, we substitute into each constraint: Flour constraint: 3(6) + 2(4) = 18 + 8 = 26 ≤ 24? NO! 26 > 24, so this constraint is violated. Oven capacity: 6 + 4 = 10 ≤ 10? YES! This constraint is satisfied. Minimum muffins: 6 ≥ 2? YES! This constraint is satisfied. Non-negativity: 6 ≥ 0 and 4 ≥ 0? YES! Both satisfied. Conclusion: nonviable because the flour constraint is violated. Choice A is correct because it correctly identifies that only the flour constraint 3x + 2y ≤ 24 is violated by the point (6, 4). Choice C says both constraints are violated, but we verified that x + y = 10 ≤ 10, so the oven capacity constraint is actually satisfied. One failure disqualifies the entire solution, but it's important to accurately identify which specific constraints are violated! When checking viability, substitute carefully: if the point is (6, 4), that means x = 6 and y = 4. Substitute those values into EVERY inequality and equation. For example, if one constraint is 3x + 2y ≤ 24, check: 3(6) + 2(4) = 18 + 8 = 26 ≤ 24? No! Do this for every single constraint. It's tedious but necessary—one missed check could mean accepting an infeasible solution!