Algebra 2 Flashcards: Modeling With Equation Inequalityconstraints

What this deck covers

This deck focuses on Modeling With Equation Inequalityconstraints, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the correct interpretation of a corner point (vertex) of a feasible region in linear constraints?

Answer: An intersection point of boundary lines that is often a candidate optimum. Corner points are where constraint boundaries intersect.

Flashcard 2: What does a constraint mean in a modeling problem?

Answer: A condition that limits allowable values of variables. Constraints establish boundaries for what's mathematically and practically possible.

Flashcard 3: What does it mean for a solution to be viable in a constraints model?

Answer: It satisfies all constraints and matches the real-world context. A viable solution meets all mathematical and practical requirements.

Flashcard 4: What does it mean for a solution to be nonviable in a constraints model?

Answer: It violates at least one constraint or is unrealistic in context. Nonviable solutions fail either mathematical or practical tests.

Flashcard 5: What is a concise way to state the solution set of a system of inequalities in context?

Answer: The set of all viable $(x,y)$ satisfying every constraint. The solution set includes all points meeting every constraint.

Flashcard 6: Identify the constraint for “The total weight is at most $100$ lb” if each $x$ weighs $8$ lb and each $y$ weighs $5$ lb.

Answer: $8x+5y \le 100$. Weight constraint limits total mass within capacity.

Flashcard 7: What is the correct interpretation of a corner point (vertex) of a feasible region in linear constraints?

Answer: An intersection point of boundary lines that is often a candidate optimum. Corner points are where constraint boundaries intersect.

Flashcard 8: Identify the system for a diet: at least $20$ g fiber ($2x+5y\ge 20$) and at most $400$ calories ($50x+80y\le 400$).

Answer: $2x+5y \ge 20$ and $50x+80y \le 400$. Multiple constraints model complex nutritional requirements simultaneously.

Flashcard 9: What inequality models “Cost $C$ must not exceed \500$”?

Answer: $C \le 500$. Cost constraints ensure spending stays within budget.

Flashcard 10: What inequality models “Revenue $R$ must be at least \500$”?

Answer: $R \ge 500$. Revenue constraints ensure minimum income requirements.

Flashcard 11: Convert the compound inequality $18 \le s \le 30$ into two inequalities.

Answer: $s \ge 18$ and $s \le 30$. Compound inequalities split into separate minimum and maximum constraints.

Flashcard 12: Identify the constraint for “At least $10$ total items” when the quantities are $x$ and $y$.

Answer: $x+y \ge 10$. Combined quantities must meet or exceed the minimum total.

Flashcard 13: Identify the constraint for “No more than $10$ total items” when the quantities are $x$ and $y$.

Answer: $x+y \le 10$. Combined quantities cannot exceed the maximum total.

Flashcard 14: What does the intersection of the graphs of two inequalities represent?

Answer: The set of points satisfying both inequalities simultaneously. The intersection shows points satisfying multiple constraints simultaneously.

Flashcard 15: Identify the inequality for “$y$ is at least twice $x$” using variables $x$ and $y$.

Answer: $y \ge 2x$. At least twice means greater than or equal to $2x$.

Flashcard 16: Identify the inequality for “$y$ is no more than twice $x$” using variables $x$ and $y$.

Answer: $y \le 2x$. No more than twice means less than or equal to $2x$.

Flashcard 17: Identify the inequality for “The difference between $x$ and $y$ is at most $5$” (single inequality form).

Answer: $|x-y| \le 5$. Absolute value constraints capture differences in either direction.

Flashcard 18: Rewrite the constraint $|x-y|\le 5$ as a system without absolute value.

Answer: $x-y \le 5$ and $x-y \ge -5$. Absolute value inequalities split into two linear constraints.

Flashcard 19: What does it mean if a system of constraints has no solution in context?

Answer: The constraints are inconsistent; there are no viable options. Conflicting constraints create an empty feasible region.

Flashcard 20: What does it mean if a system of constraints has infinitely many solutions in context?

Answer: There are many viable options; the feasible region has area/length. Multiple solutions form a region rather than discrete points.

Flashcard 21: Identify the constraint for “At least $4$ units of $x$” when $x$ must be an integer count.

Answer: $x \ge 4$ and $x \in \mathbb{Z}$. Integer constraints model discrete counting situations.

Flashcard 22: What is the meaning of a point outside the feasible region in a modeling context?

Answer: A nonviable option because it breaks one or more constraints. Points outside violate constraints and are impractical solutions.

Flashcard 23: What additional constraint models that $x$ and $y$ count items and cannot be fractional?

Answer: $x \in \mathbb{Z}$ and $y \in \mathbb{Z}$ (often with $x,y\ge 0$). Discrete quantities require integer domain restrictions.

Flashcard 24: A class requires at least $18$ students and at most $30$. What compound inequality models this for $s$?

Answer: $18 \le s \le 30$. Compound inequalities capture both minimum and maximum bounds.

Flashcard 25: Which check determines whether $(x,y)$ is a solution to a system of inequalities?

Answer: Substitute into every inequality and verify all are true. A solution must satisfy every constraint in the system.

Flashcard 26: Find whether $(4,5)$ satisfies $2x+y<13$.

Answer: Nonviable; $2(4)+5=13$ is not less than $13$. Strict inequality requires the value be less than, not equal to $13$.

Flashcard 27: Find whether $(3,2)$ satisfies the system $x\ge 0$, $y\ge 0$, and $x+y\le 4$.

Answer: Nonviable; $3+2=5$ violates $x+y\le 4$. The sum $3+2=5$ exceeds the maximum allowed total of $4$.

Flashcard 28: Find whether $(1,3)$ satisfies the system $x\ge 0$, $y\ge 0$, and $x+y\le 4$.

Answer: Viable; $1+3=4$ and both are nonnegative. All constraints are satisfied: $1 \ge 0$, $3 \ge 0$, and $4 \le 4$.

Flashcard 29: What is the meaning of the boundary line being included in the solution set?

Answer: The inequality uses $\le$ or $\ge$, so equality is allowed. Non-strict inequalities include the boundary as valid solutions.

Flashcard 30: What inequality symbol models “at least” in a constraint statement?

Answer: $\ge$. Greater than or equal to includes the minimum value.