HSA.CED.A.3: Representing Constraints and Judging Viable Solutions
In plain English: HSA.CED.A.3 is the Common Core algebra standard that asks students to represent the constraints in a situation with equations, inequalities, or systems of equations and inequalities, and to decide whether a solution is viable or nonviable in context. A viable solution meets every constraint and makes sense, such as a whole number of buses. It is usually taught in Algebra I.
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Common Core State Standards for Mathematics · Domain: Creating Equations (CED) · Cluster: Create equations that describe numbers or relationships Also written as HSA-CED.A.3 or A-CED.3 · Official standard
In this lesson, students turn the limits in a real situation (a budget, a number of hours, a supply of materials, a required minimum) into equations and inequalities, and they combine several constraints into a system. The second half of the standard is just as important: once they have a possible solution, students decide whether it is viable, meaning it satisfies every constraint and makes sense in the context, or nonviable.
Students work with four kinds of models: a single inequality, a single equation, a system of equations and a system of inequalities. They test candidate points by substitution, read a graphed solution region, and explain why answers such as 2.5 bouquets, -3 hours or 5.5 tents cannot be reported as real options even when they satisfy the algebra.
Learning Objectives
By the end of this lesson, students will be able to:
Identify the quantities and the limits in a context and write each limit as an equation or inequality
Combine several constraints, including nonnegativity, into a system of equations or a system of inequalities
Test a candidate solution by substituting it into every constraint
Classify a solution as viable or nonviable and explain which constraint or which feature of the context it fails
Use a graph of a system of inequalities to describe the set of viable options
Prior Knowledge Required
Students should already be comfortable with:
Writing and solving word problems that lead to linear equations and inequalities 7.EE.B.4
Solving systems of two linear equations in two variables 8.EE.C.8
Writing equations in two variables from a context HSA.CED.A.2
Translating phrases such as "at most," "at least" and "no more than" into inequality symbols
Show the prompt and let students work alone for 3 minutes, then compare answers with a partner.
Warm-Up Prompt
"A club has $240 to spend on food for a party. Pizzas cost $12 each and orders of wings cost $8 each. Give two different orders the club could afford and one order it could not afford."
Record several answers on the board, such as (10 pizzas, 15 wings) and (12 pizzas, 15 wings). Ask how students checked each one. Guide the class to the rule they were all using, 12p + 8w ≤ 240, and name it a constraint. Then ask: "Is 4.5 pizzas and 20 wings affordable?" It satisfies the inequality (54 + 160 = 214), but a pizzeria sells whole pizzas, which sets up the idea that a solution must also make sense in context.
Direct Instruction20 minutes
Teach a four-step routine for modeling with constraints:
Define the variables: Name each quantity you can choose, with its unit, for example "x = hours tutoring per week."
Write one constraint per limit: Use = for an exact requirement ("uses exactly 100 roses") and an inequality for a limit or a minimum ("at most 20 hours," "at least $288"). Add x ≥ 0 and y ≥ 0 when quantities cannot be negative.
Find or test solutions: Solve the system, read the graph, or substitute a candidate point into every constraint.
Judge viability: A solution is viable only if it satisfies every constraint and fits the context (whole numbers when items are counted, no negative amounts, realistic values). If it is nonviable, say which condition fails.
Work through one example of each type of model:
Single inequality
"The club has at most $240 for pizzas at $12 and wings at $8. Is 10 pizzas and 15 wings viable? What about 12 pizzas and 15 wings?"
"A florist uses exactly 100 roses to make small bouquets of 5 roses and large bouquets of 12 roses. Which combinations are possible?"
Equation: 5s + 12l = 100; viable: (20, 0) and (8, 5); (14, 2.5) fits the equation but is nonviable
System of equations
"A theater sold 300 tickets. Adult tickets cost $12, student tickets cost $7, and ticket sales were $2,900. How many of each were sold?"
Equation: a + s = 300 and 12a + 7s = 2900; a = 160, s = 140, viable
System of inequalities
"A student can work at most 20 hours a week, tutoring at $18 per hour and lifeguarding at $12 per hour, and wants to earn at least $288."
Equation: x + y ≤ 20, 18x + 12y ≥ 288, x ≥ 0, y ≥ 0; (14, 4) viable, (6, 10) and (12, 10) nonviable
A nonviable solution of a system
"A scout troop has exactly 10 tents: 6-person tents and 4-person tents. Can the tents hold exactly 51 campers with every spot filled?"
Equation: x + y = 10 and 6x + 4y = 51 give x = 5.5, y = 4.5, nonviable
Graph the work-hours example with the class using Diagram 1. Shade the region where both inequalities hold, and test the three marked points by substitution before looking at the graph. Emphasize that the corner point (8, 12) comes from solving the two boundary equations together, and that it is viable because both constraints include their boundary (≤ and ≥). Close the example with the tent problem: the system has a solution, but half a tent cannot be used, so the answer is "no combination works," not "5.5 tents."
Guided Practice15 minutes
Pairs work on two scenarios. (1) A robotics club can build at most 60 items: keychains earn $3 profit each and bracelets earn $2 profit each, and the club wants at least $150 in profit. Write the system and test (40, 15), (20, 30) and (50, 12). (2) A school buys 40 calculators, some scientific at $15 and some graphing at $110, and spends $2,120. Write and solve the system. Circulate and ask each pair: "Which constraint does this point break?" and "Did you include the constraints that the numbers cannot be negative?" Debrief (50, 12), which meets the profit goal ($174) but uses 62 items.
Independent Practice15 minutes
Students work alone on a food problem and a transportation problem. Food, with nutrition and cost constraints (the standard's own example): a smoothie uses y cups of yogurt (10 g protein, 150 calories and $0.80 per cup) and f cups of fruit (1 g protein, 100 calories and $0.50 per cup) and must have at least 12 g of protein, at most 400 calories and a cost of at most $2.00. Transportation: a school needs seats for 250 people using at most 7 vehicles, buses with 45 seats and vans with 12 seats. For each, students write the system, test two options of their own, and classify each as viable or nonviable with a reason. Point out that fractional cups are allowed in the first problem but fractional buses are not in the second.
Closure5-10 minutes
Exit ticket: "A camp needs at least 10 staff members each day. Counselors cost $50 per day and lifeguards cost $80 per day, and the daily budget is at most $700. (a) Write the system. (b) Is 3 counselors and 7 lifeguards viable? (c) Is 7 counselors and 3 lifeguards viable?" A correct ticket shows x + y ≥ 10 and 50x + 80y ≤ 700, rejects (3, 7) because it costs $710, and accepts (7, 3) at $590.
Differentiation Strategies
For Struggling Students
Give a constraint table with columns "Limit in words," "Symbol" and "Inequality" so students translate one limit at a time
Start with a single inequality and a checklist for testing points before moving to systems
Provide the graph of the system already drawn and ask students only to test and classify points
Keep a class anchor chart of phrases: "at most" means ≤, "at least" means ≥, "exactly" means =
For Advanced Students
Find every whole-number viable option in the bus problem and explain how you know the list is complete
Add a third constraint to the work-hours problem (for example, at least 4 hours of lifeguarding) and describe how the viable region changes
Find the viable option that maximizes earnings or profit by testing the corner points of the region, as a preview of linear programming
Assessment Guidance
What to Look For
Look for two separate skills. First, each constraint should match its limit in words, including the direction of the inequality and whether the boundary is included. Second, students should justify every viability decision by naming the constraint or the feature of the context that a point fails. A student who writes "5.5 tents" or reports a point outside the shaded region has solved the algebra but has not yet met the standard. Also check that students include nonnegativity constraints when they matter.
02
Classroom Activities
3 Activities
1
Plan the Class Trip
25 minGroups of 3-4
Groups receive a planning sheet for a class trip with several limits. They write every constraint, propose three plans, and present one plan to the class with evidence that it is viable.
Planning Sheet
At least 54 students and chaperones must travel
Minivans seat 7 people and cost $90 per day; large vans seat 12 people and cost $150 per day
The transportation budget is at most $750
Only 6 drivers are available, so at most 6 vehicles can be used
Procedure
Define m = number of minivans and v = number of large vans, then write 7m + 12v ≥ 54, 90m + 150v ≤ 750, m + v ≤ 6, m ≥ 0 and v ≥ 0
Propose three plans and test each against every constraint (for example, 2 minivans and 3 large vans seat only 50 people, so that plan is nonviable, while 1 minivan and 4 large vans seat 55 people for $690, which is viable)
Present the cheapest viable plan the group found and explain why each rejected plan fails (the whole-number options that work are 1 minivan and 4 large vans, 3 and 3, and 0 and 5; the cheapest is $690)
Modification for Distance Learning
Share the planning sheet as an online document. Each breakout group fills in a table of plans with a column for each constraint and a final "viable or nonviable" column.
2
Viable or Not? Card Sort
15 minPairs
Each pair gets one scenario with its system and 8 candidate solution cards. Pairs sort the cards into "viable" and "nonviable" piles and write the reason on the back of every nonviable card.
Sample Scenario and Cards
Scenario: A bakery has 20 cups of flour. Cookies need 2 cups per batch and brownies need 3 cups per batch, and the bakery must make at least 4 batches of cookies. System: 2c + 3b ≤ 20, c ≥ 4, b ≥ 0.
Viable cards: (4, 4), (7, 2), (10, 0)
Nonviable cards: (3, 4) fails c ≥ 4; (6, 3) uses 21 cups; (4, 2.5) is half a batch; (4, -1) is negative; (8, 2) uses 22 cups
Discussion Questions
Which cards satisfy every inequality but still fail the context?
Could half a batch ever make sense? What would have to change about the situation?
What is the greatest number of brownie batches possible, and how do you know?
3
Design a Constraint Problem
20 minIndividual then share
Students write their own constraint problem from a context they know (a part-time job, a sports team budget, a garden, a fundraiser). The problem must have at least two constraints, one viable option and one nonviable option.
Requirements
Variables defined with units, and at least two constraints written as equations or inequalities
A graph of the system or a table of tested options
One viable option and one nonviable option, each with a written justification
Partner Swap Variation
Students trade problems with a partner, who writes the constraints independently and tests the two options. Pairs compare their systems and resolve any differences before turning in both versions.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A System of Inequalities and Its Viable Region
The constraints x + y ≤ 20 and 18x + 12y ≥ 288, with x ≥ 0 and y ≥ 0, drawn to scale. Both boundary lines are solid because both inequalities include equality. Points in the shaded triangle, including its edges, are viable. (6, 10) fails the earnings constraint and (12, 10) fails the hours constraint.
Diagram 2: Deciding Whether a Solution Is Viable
A two-check routine. Check 1 is algebraic: the point must satisfy every constraint. Check 2 is about the context: the values must be possible in the real situation. A point that fails either check is nonviable, and students should say which check it fails.
04
Homework Assignment
~30 min
HSA.CED.A.3 Homework: Constraints and Viable Options
Directions: For each problem, (a) define your variables with units, (b) write every constraint, including any that keep quantities from being negative, (c) answer the question, and (d) state whether each option is viable or nonviable and explain why.
Part 1: Equations and Inequalities as Constraints (Problems 1-3)
A food truck buys packs of hamburger buns for $3 and packs of hot dog buns for $2.50, and it can spend at most $45. Write the constraint. Is buying 10 packs of hamburger buns and 6 packs of hot dog buns viable? What about 8 packs and 9 packs?
A school spent exactly $1,860 on 36 chairs. Plastic chairs cost $35 and padded chairs cost $65. Write a system of equations and solve it. Is your solution viable?
A farmer builds a rectangular pen against a barn, so fencing is needed on only three sides: two widths w and one length l. She has exactly 36 feet of fencing, and the barn wall is 24 feet long, so the length can be at most 24 feet. Write the constraints. Is a width of 5 feet viable? A width of 8 feet? What is the smallest viable width?
Part 2: Systems of Inequalities (Problems 4-6)
A breakfast bowl uses o cups of dry oats and m cups of milk. Each cup of oats has 10 g of protein and 300 calories and costs $0.30, and each cup of milk has 8 g of protein and 100 calories and costs $0.70. The bowl must have at least 14 g of protein and at most 450 calories, and it can cost at most $1.00. Write the system. Classify each option (o, m): (1, 1), (0.5, 1), (1, 2) and (0.5, 1.5).
A school store can stock at most 80 items in total. Each notebook earns $2 profit and each water bottle earns $5 profit, and the store wants at least $250 in profit. Write the system and graph it with notebooks n on the horizontal axis. Find the corner point where the two boundary lines meet, and classify (30, 40) and (60, 15).
A school must seat 200 people using at most 6 vehicles: buses that seat 40 and vans that seat 14. (a) A student writes b + v = 6 and 40b + 14v = 200. Solve this system and explain why the solution is nonviable. (b) Write the constraints as a system of inequalities instead and list every viable whole-number option.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Constraints
Every limit written correctly, including nonnegativity
Most constraints correct, one symbol or term wrong
Constraints missing or incorrect
Solving and Testing
Solutions and substitutions shown and correct
Method shown, one arithmetic error
No work shown
Viability Decision
Each option classified with the constraint or context reason named
Classified without a clear reason
Not classified
Graph (Problem 5)
Labeled axes, correct boundaries, correct region
Region or one boundary incorrect
No graph
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A caterer can spend at most $500 on chicken at $4 per pound and beef at $6 per pound. If c is pounds of chicken and b is pounds of beef, which inequality represents the budget?
Answer: C
"At most $500" means the cost can equal $500 or be less, so the symbol is ≤, and the cost is 4c + 6b. Choice A reverses the direction, which would describe a minimum spending amount. Choice B attaches each price to the wrong meat. Choice D charges $10 for every pound.
Question 2 of 20 · Multiple Choice
Using the budget 4c + 6b ≤ 500 from the previous question, which purchase is viable?
Answer: B
Check (50, 50): 4(50) + 6(50) = 200 + 300 = 500, which is at most $500, so it is viable. The other choices cost $540, $520 and $530. Choice D is a common trap: 125 pounds of chicken alone uses the whole $500, so any beef goes over budget.
Question 3 of 20 · Multiple Choice
A jar holds 30 coins, all dimes and quarters, worth $4.50 in total. If d is the number of dimes and q is the number of quarters, which system represents the constraints?
Answer: D
One equation counts coins (d + q = 30) and the other adds their values in dollars (0.10d + 0.25q = 4.50). Choice A swaps the count and the value. Choice B mixes cents on the left with dollars on the right. Choice C gives each coin the other coin's value.
Question 4 of 20 · Multiple Choice
Solve the system from the previous question. How many quarters are in the jar?
Answer: A
Substitute d = 30 - q: 0.10(30 - q) + 0.25q = 4.50, so 3 + 0.15q = 4.50 and q = 10. Then d = 20. Check: 20 dimes are $2.00 and 10 quarters are $2.50, for $4.50. Choice B is the number of dimes, not quarters.
Question 5 of 20 · Multiple Choice
A student models the number of tables to rent for an event and finds that the system's solution is 7.5 tables. What is the best response?
Answer: A
Half a table cannot be rented, so 7.5 is nonviable in this context. The student should test whole-number options such as 7 and 8 in every constraint and report one that works. Choice D is a rule that can fail: rounding down may break a "seat at least" constraint. Choice C confuses a nonviable solution with no solution.
Question 6 of 20 · Multiple Choice
Which point is a solution of the system x + y ≤ 10, 2x + y ≥ 12, x ≥ 0, y ≥ 0?
Answer: C
Check (6, 2): 6 + 2 = 8 ≤ 10 and 2(6) + 2 = 14 ≥ 12, so both inequalities hold. (2, 6) gives 2(2) + 6 = 10, which is less than 12. (8, 4) gives 8 + 4 = 12, which is more than 10. (3, 3) gives 9 for the second inequality.
Question 7 of 20 · Multiple Choice
A school rule requires at least 1 chaperone for every 8 students on a trip. If c is the number of chaperones and s is the number of students, which inequality represents the rule?
Answer: C
Each chaperone can cover 8 students, so c chaperones cover 8c students, and that must be at least the number of students: 8c ≥ s. Test it: 3 chaperones and 24 students gives 24 ≥ 24, allowed. Choice A is a common reversal error; it would require 8 chaperones for every student.
Question 8 of 20 · Multiple Choice
Why do many constraint systems include x ≥ 0 and y ≥ 0?
Answer: A
Nonnegativity constraints come from the context: you cannot work -3 hours or buy -2 items. Choice D is wrong because x ≥ 0 still allows values such as 2.5; a whole-number requirement comes from the context and must be checked separately.
Question 9 of 20 · Multiple Choice
A system that models hours worked at two jobs has the solution (-2, 14). How should this solution be interpreted?
Answer: B
The point satisfies the equations, but a negative number of hours has no meaning, so it is nonviable. Choice A ignores the context. Choice C changes the solution: (0, 14) is a different point that would have to be tested in every constraint on its own.
Question 10 of 20 · Multiple Choice
A constraint is 3x + 5y < 60. The point (10, 6) lies on the boundary line 3x + 5y = 60. Is (10, 6) a solution of the constraint?
Answer: D
3(10) + 5(6) = 60, and 60 is not less than 60, so the point does not satisfy the strict inequality. On a graph, this boundary would be dashed. Choice A would be true only for ≤ or ≥. Choice C is false: the point is on the line.
Question 11 of 20 · Multiple Choice
A baker must use exactly 60 eggs, making small cakes that need 4 eggs and large cakes that need 6 eggs. Which option is viable?
Answer: D
Check (9, 4): 4(9) + 6(4) = 36 + 24 = 60 eggs, with whole cakes, so it is viable. Choice A also gives 30 + 30 = 60, but half a cake is not a real option here, so it is nonviable. Choice B uses 62 eggs and Choice C uses only 48, so neither uses exactly 60.
Question 12 of 20 · Multiple Choice
A workshop has 80 hours of assembly time. Each chair takes 2 hours and each table takes 5 hours. If c is the number of chairs and t is the number of tables, which inequality represents the time constraint?
Answer: A
Chairs use 2c hours and tables use 5t hours, and the total cannot be more than 80: 2c + 5t ≤ 80. Choice B switches the times. Choice C would require at least 80 hours of work. Choice D assumes each item takes 7 hours.
Question 13 of 20 · Multiple Choice
A club can make at most 50 items and wants at least $180 profit: c + s ≤ 50 and 4c + 3s ≥ 180, where c is candles and s is bars of soap. If the club makes no soap, what is the fewest candles it can make?
Answer: B
With s = 0, the profit constraint is 4c ≥ 180, so c ≥ 45, and c = 45 also meets c + s ≤ 50. Choice A is the most items allowed, not the fewest needed. Choice C comes from dividing 180 by 3, the soap profit, and would also break the 50-item limit. Choice D divides 180 by 5.
Question 14 of 20 · Multiple Choice
A concert sold 250 tickets for $3,400. Floor tickets cost $16 and balcony tickets cost $10. How many floor tickets were sold?
Answer: C
Solve f + b = 250 and 16f + 10b = 3400. Substituting b = 250 - f gives 16f + 2500 - 10f = 3400, so 6f = 900 and f = 150. Then b = 100. Choice A is the number of balcony tickets. Choice D comes from dividing 3,400 by 16 and rounding, which ignores the balcony tickets.
Question 15 of 20 · Short Answer
A farmer has at most 120 acres to plant with corn and soybeans. Corn costs $300 per acre and soybeans cost $200 per acre, and the budget is at most $30,000. Write the system of constraints. Is 80 acres of corn and 40 acres of soybeans viable? Is 60 acres of each viable?
System: c + s ≤ 120, 300c + 200s ≤ 30,000, c ≥ 0, s ≥ 0 (80, 40): 120 acres is allowed, but the cost is 24,000 + 8,000 = $32,000, over budget, so it is nonviable. (60, 60): 120 acres and 18,000 + 12,000 = $30,000, exactly the budget, so it is viable.
Question 16 of 20 · Short Answer
On a school night, Sam wants to study at least 3 hours, and studying plus gaming can total at most 6 hours. Let s be hours of studying and g be hours of gaming. Write the constraints, and classify (2, 3) and (4, 1.5).
Constraints: s ≥ 3, s + g ≤ 6, g ≥ 0 (2, 3) is nonviable because 2 hours of study is less than 3. (4, 1.5) is viable: 4 ≥ 3 and 4 + 1.5 = 5.5 ≤ 6. Fractions of an hour make sense here, so 1.5 hours is fine.
Question 17 of 20 · Short Answer
At a carnival, rides cost 3 tickets and games cost 2 tickets. Mia has 40 tickets and wants to go on at least 6 rides. Write the constraints. If she goes on 8 rides, how many games can she play at most? Is 12 rides and 3 games viable?
Constraints: 3r + 2g ≤ 40, r ≥ 6, g ≥ 0 With r = 8: 24 + 2g ≤ 40, so g ≤ 8. She can play at most 8 games. For (12, 3): 36 + 6 = 42 tickets, more than 40, so it is nonviable.
Question 18 of 20 · Short Answer
In your own words, explain the difference between a solution of a system and a viable solution. Give an example.
A solution of a system is any pair of numbers that makes every equation or inequality true. A viable solution also has to make sense in the real situation. For example, 8 boxes that must hold exactly 45 books, using boxes of 6 books and boxes of 4 books, give x + y = 8 and 6x + 4y = 45, whose solution is x = 6.5 and y = 1.5. That pair solves the system, but half a box cannot be filled, so it is nonviable.
Question 19 of 20 · Short Answer
A craft club has 30 feet of ribbon. Each wreath needs 3 feet and each bow needs 2 feet, and the club must make at least 6 wreaths. Write the constraints. What is the greatest number of bows it can make?
Constraints: 3w + 2b ≤ 30, w ≥ 6, b ≥ 0 The most ribbon is left for bows when w = 6: 18 + 2b ≤ 30, so 2b ≤ 12 and b ≤ 6. The club can make at most 6 bows, using exactly 30 feet.
Question 20 of 20 · Short Answer
A tournament needs at least 12 referees each day. Head referees are paid $40 per day and assistant referees $25 per day, and the daily budget is at most $400. Is 7 head referees and 5 assistants viable? Is 4 head referees and 8 assistants viable?
Constraints: h + a ≥ 12, 40h + 25a ≤ 400, h ≥ 0, a ≥ 0 (7, 5): 12 referees, but 280 + 125 = $405, over budget, so it is nonviable. (4, 8): 12 referees and 160 + 200 = $360, so it is viable.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What is a constraint?
A constraint is a limit or requirement in a situation, written as an equation or an inequality. "The budget is at most $240" becomes 12p + 8w ≤ 240. "The florist uses exactly 100 roses" becomes 5s + 12l = 100. A real problem usually has several constraints at once, which together form a system.
What is the difference between a viable and a nonviable solution?
A viable solution satisfies every constraint and also makes sense in the context. A nonviable solution fails at least one of those. It might break a constraint, like a plan that goes over budget, or it might satisfy the algebra but be impossible in real life, like 2.5 bouquets or -3 hours. When students call a solution nonviable, they should say exactly why.
When should a constraint be an equation instead of an inequality?
Use an equation when the situation requires an exact amount: "exactly 300 tickets," "uses all 36 feet of fencing," "total sales were $2,900." Use an inequality when there is a limit or a minimum: "at most," "no more than," "at least," "a minimum of." Many problems mix both, such as a fixed number of vehicles with a minimum number of seats.
Why include constraints like x ≥ 0 and y ≥ 0?
Most quantities in modeling problems, such as hours, items, acres and people, cannot be negative. Without x ≥ 0 and y ≥ 0, the graph of a system can include points with negative values that satisfy the other inequalities but have no meaning. Writing these constraints explicitly keeps the viable region inside the first quadrant.
If a solution is a decimal, should students just round it?
Not without checking. If the quantity must be a whole number, the decimal solution is nonviable, and students should test nearby whole-number options in every constraint. Rounding can break a constraint: in the bus problem on this page, 5 buses and 2 vans seat 249 people, one short of the 250 required. Sometimes no whole-number option satisfies the constraints, and that is a valid answer to report.
Do decimal answers always make a solution nonviable?
No. It depends on the quantity. Cups of fruit, hours, pounds and gallons can take fractional values, so a solution such as 1.5 hours is fine. Buses, tickets, people and bouquets are counted in whole numbers. Students should decide which kind each variable is when they define it.
How is a graph used to find viable options?
Graph the boundary line of each constraint, shade the side that satisfies it, and look for the region where all the shading overlaps. Use a solid line when the inequality includes equality (≤ or ≥) and a dashed line when it does not (< or >). Every point in the overlap satisfies the system. Students still need to check the context, for example by keeping only whole-number points when items are counted.
What mistakes do students often make on this standard?
Reversing a rate constraint, such as writing g ≥ 6t instead of 6t ≥ g when t tables that seat 6 each must hold g guests
Choosing ≤ when the context says "at least"
Leaving out nonnegativity constraints
Reporting a decimal or negative solution as the answer without comment
Testing a point in only one of the constraints
Is HSA.CED.A.3 tested on the SAT?
The skills are. The Algebra domain of the digital SAT includes writing linear inequalities in one or two variables from a context, systems of two linear equations in two variables, and questions that ask which values satisfy a set of conditions. Practicing translating constraints and testing points prepares students for those question types.
What comes after HSA.CED.A.3?
Systems of inequalities with a viable region are the starting point for linear programming, where students find the best option (for example, the most profit) by testing the corners of the region. The same idea of constraints and feasible options appears in economics, engineering and computer science. Within the Common Core, HSA.REI.C.6 and HSA.REI.D.12 develop the solving and graphing skills that this standard applies.
07
Related Standards
5 standards
These standards connect to HSA.CED.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.EE.B.4Prerequisite
Solve word problems by writing and solving simple equations and inequalities
