Constraints and viable solutions (SAT). Game on.

Constraints and viable solutions (SAT) is a grade 9 math skill aligned to Common Core standard HSA.CED.A.3: 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. Below are 8 practice questions with answers and step-by-step explanations, drawn from the 10 constraints and viable solutions (sat) problems our math games drill.

CCSS HSA.CED.A.310 questions in the bank
Kickoff

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Warm-upeasy

A concert sells ss student tickets and aa adult tickets. There are 22 times as many student tickets as adult tickets, and 2727 tickets are sold in all. Which system of equations represents this situation?

There are 22 times as many student tickets as adult tickets, so s=2a.s = 2a\textsf{.} In all there are 2727 tickets, so s+a=27.s + a = 27\textsf{.}

Mid-gameeasy

A student can spend at most $48\text{\char36}48 on adult tickets at $12\text{\char36}12 each and child tickets at $8\text{\char36}8 each. She wants at least 22 child tickets. Let aa be the number of adult tickets and cc the number of child tickets. Which pair (a,c)(a, c) is a possible purchase?

Check both constraints: 12a+8c≤4812a + 8c \le 48 and c≥2.c \ge 2\textsf{.} The pair (2,3)(2, 3) costs 12(2)+8(3)=4812(2) + 8(3) = 48 and has 33 child tickets, so both constraints are met.

Mid-gameeasy

A student can spend at most $24\text{\char36}24 on sandwiches at $5\text{\char36}5 each and drinks at $3\text{\char36}3 each, and wants at least 22 drinks. Let ss be the number of sandwiches and dd the number of drinks. Which pair (s,d)(s, d) is a possible order?

Check both constraints: 5s+3d≤245s + 3d \le 24 and d≥2.d \ge 2\textsf{.} The pair (3,3)(3, 3) costs 5(3)+3(3)=245(3) + 3(3) = 24 and has 33 drinks, so both constraints are met.

Mid-gameeasy

A club has $85\text{\char36}85 to pay student helpers $20\text{\char36}20 each. What is the greatest number of helpers the club can hire?

85÷20=4.25.85 \div 20 = 4.25\textsf{.} A helper cannot be hired in a fraction, and 55 helpers would cost $100.\text{\char36}100\textsf{.} The greatest whole number of helpers the club can pay is 4.4\textsf{.}

Mid-gameeasy

A student needs 77 ounces of chocolate. Each candy bar is 44 ounces. What is the least number of candy bars the student should buy?

7÷4=1.75.7 \div 4 = 1.75\textsf{.} A bar cannot be bought in a fraction, and 11 bar is only 44 ounces. Two bars are 88 ounces, which meets the 77 ounces needed.

Mid-gameeasy

A club can spend at most $30\text{\char36}30 on notebooks at $6\text{\char36}6 each and pens at $4\text{\char36}4 each, and must buy at least 22 notebooks. Let nn be the number of notebooks and pp the number of pens. Which pair (n,p)(n, p) is a possible purchase?

Check both constraints: 6n+4p≤306n + 4p \le 30 and n≥2.n \ge 2\textsf{.} The pair (3,3)(3, 3) costs 6(3)+4(3)=306(3) + 4(3) = 30 and has 33 notebooks, so both constraints are met.

Mid-gamemedium

A band can spend at most $60\text{\char36}60 on shirts that cost $12\text{\char36}12 each and hats that cost $8\text{\char36}8 each. The band wants at least 33 hats. Let ss be the number of shirts and hh the number of hats. Which system represents these constraints?

At most $60\text{\char36}60 gives 12s+8h≤60.12s + 8h \le 60\textsf{.} At least 33 hats gives h≥3.h \ge 3\textsf{.}

Buzzer beatermedium

A club sells student tickets for $4\text{\char36}4 and adult tickets for $6.\text{\char36}6\textsf{.} The ticket money totals $22.\text{\char36}22\textsf{.} Let xx be the number of student tickets and yy the number of adult tickets. Which pair (x,y)(x, y) is a possible sale?

The money equation is 4x+6y=22.4x + 6y = 22\textsf{.} Ticket counts must be whole numbers that are zero or greater. The pair (4,1)(4, 1) gives 4(4)+6(1)=224(4) + 6(1) = 22 and both counts are whole and nonnegative.

Overtime

2 more. Played, not assigned.

The rest of the bank lives inside live head-to-head matches — the placement test finds the right starting difficulty.

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Tests that cover constraints and viable solutions (sat):
NAEPACTSBACLEAP 2025NC EOG/EOC
Reporting domains for this skill
ACT All grades: Integrating Essential SkillsSTAAR Algebra I: Linear Functions, Equations, and InequalitiesMAP Growth Math 6+: Operations and Algebraic ThinkingSBAC High School: Concepts and ProceduresNAEP Grade 12: AlgebraACT Aspire All grades: Preparing for Higher Math
More Grade 9 skills
Compound inequality situationsDifference of squares factoringFind score from meanInterpreting data displaysParallel and perpendicular slopes from slope-intercept formCommon factor group size (ACT)Discriminant analysisExpected value from probability (ACT)All Grade 9 practice →

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