Algebra 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.

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 does the constraint "at most $50$" translate to using an inequality?

Answer: $\le 50$. Maximum possible value allowed.

Flashcard 2: What does the constraint "exactly $50$" translate to as an equation?

Answer: $= 50$. Precise value with no variation allowed.

Flashcard 3: What does the constraint "between $3$ and $7$, inclusive" translate to?

Answer: $3 \le x \le 7$. Range includes both endpoint values.

Flashcard 4: Identify the constraint for "$x$ is at least twice $y$."

Answer: $x \ge 2y$. Sets lower bound as double the other variable.

Flashcard 5: What is the standard first step to test whether $(a,b)$ satisfies an inequality constraint?

Answer: Substitute $x=a$ and $y=b$ and check if it is true. Direct substitution verifies constraint satisfaction.

Flashcard 6: What does a system of inequalities represent in a modeling problem?

Answer: All constraints together; solutions satisfy every inequality. Multiple restrictions that must all hold simultaneously.

Flashcard 7: Identify the constraint for "$x$ exceeds $y$ by at least $3$."

Answer: $x - y \ge 3$. Difference must meet the minimum threshold.

Flashcard 8: What inequality represents "$x$ is nonnegative" in a modeling context?

Answer: $x \ge 0$. Cannot be negative in most contexts.

Flashcard 9: What does a system of inequalities represent in a modeling problem?

Answer: All constraints together; solutions satisfy every inequality. Multiple restrictions that must all hold simultaneously.

Flashcard 10: What does the constraint "exactly $50$" translate to as an equation?

Answer: $= 50$. Precise value with no variation allowed.

Flashcard 11: Identify whether $(4,0)$ satisfies the constraint $2x + y \le 7$.

Answer: Nonviable, because $2(4)+0=8>7$. Substitution shows constraint violation.

Flashcard 12: Which line style is used for $\le$ or $\ge$ when graphing an inequality?

Answer: A solid boundary line. Points on the line satisfy the constraint.

Flashcard 13: What does the constraint "no more than $50$" translate to using an inequality?

Answer: $\le 50$. Upper limit that cannot be exceeded.

Flashcard 14: Identify the viability of $(x,y)=(-1,5)$ for the system $x\ge 0$, $y\ge 0$, $x+y\le 10$.

Answer: Nonviable, because $x=-1$ violates $x\ge 0$. Negative value violates non-negativity constraint.

Flashcard 15: What is the meaning of the intersection point of two boundary lines in a constraints model?

Answer: A point that satisfies both equalities at the same time. Corner where two constraints meet exactly.

Flashcard 16: What does the inequality symbol $<$ mean in a constraint?

Answer: $<$ means "strictly less than.". Excludes the boundary value itself.

Flashcard 17: In a system, what does it mean if a point satisfies all but one inequality?

Answer: It is nonviable because one violated constraint is enough. Any constraint violation makes solution invalid.

Flashcard 18: What two inequalities are equivalent to the constraint $|x-y| \le 3$?

Answer: $-3 \le x-y \le 3$. Absolute value inequalities split into compound form.

Flashcard 19: What inequality models "no more than $500$ calories" if foods give $120x$ and $200y$ calories?

Answer: $120x + 200y \le 500$. Total calorie intake has an upper bound.

Flashcard 20: Identify the constraint for "$x$ is at most $5$ less than $y$."

Answer: $x \ge y - 5$. Maximum amount one can fall below another.

Flashcard 21: What does the constraint "at most $50$" translate to using an inequality?

Answer: $\le 50$. Maximum possible value allowed.

Flashcard 22: What does the inequality symbol $<$ mean in a constraint?

Answer: $<$ means "strictly less than.". Excludes the boundary value itself.

Flashcard 23: What is the meaning of the word "constraint" in a modeling problem?

Answer: A condition that limits the possible solutions. Restrictions define what's possible in the model.

Flashcard 24: What inequality models "total items is at most $30$" using $x$ and $y$?

Answer: $x + y \le 30$. Combined quantities stay within the limit.

Flashcard 25: Find the correct constraint for "$x$ is no more than $5$ more than $y$."

Answer: $x \le y + 5$. Upper limit on how much one exceeds another.

Flashcard 26: What does the constraint "no more than $50$" translate to using an inequality?

Answer: $\le 50$. Upper limit that cannot be exceeded.

Flashcard 27: What does the solution set of a system of inequalities correspond to on a graph?

Answer: The overlap (intersection) of all shaded regions. Only where all constraints are satisfied together.

Flashcard 28: What does the boundary line of an inequality represent on a graph?

Answer: Where the inequality becomes an equality. The edge between satisfied and unsatisfied regions.

Flashcard 29: Identify the constraint for "$x$ is within $3$ of $y$" using an absolute value inequality.

Answer: $|x-y| \le 3$. Distance between values stays within bounds.

Flashcard 30: Which line style is used for $<$ or $>$ when graphing an inequality?

Answer: A dashed boundary line. Points on the line don't satisfy the constraint.

Flashcard 31: Identify the correct inequality for a budget of $120 with costs $5x$ and $8y$.

Answer: $5x + 8y \le 120$. Total cost cannot exceed the budget limit.

Flashcard 32: What does the constraint "no less than $50$" translate to using an inequality?

Answer: $\ge 50$. Minimum threshold that must be met.

Flashcard 33: What inequality models "at least $8$ hours total" if activity A is $x$ hours and B is $y$ hours?

Answer: $x + y \ge 8$. Combined time must meet the minimum.

Flashcard 34: What is the feasible region for a system of constraints?

Answer: The set of all points satisfying every constraint. All valid solutions within constraint boundaries.

Flashcard 35: What system models "at least $10$ items total" and "budget at most $60" with costs $4x$ and $6y$?

Answer: $x+y \ge 10,\ 4x+6y \le 60$. Combined quantity and budget constraints.

Flashcard 36: Identify the inequality for "You can make no more than $20$ units total" with $x$ and $y$ units.

Answer: $x + y \le 20$. Maximum capacity limitation constraint.

Flashcard 37: What inequality models "use at most $60$ minutes" if task 1 takes $5x$ minutes and task 2 takes $8y$ minutes?

Answer: $5x + 8y \le 60$. Total time cannot exceed the available time.

Flashcard 38: What does it mean for a solution to be "viable" in a modeling context?

Answer: It satisfies all constraints and makes sense in context. Realistic and mathematically valid solution.

Flashcard 39: What does the constraint $x \ge 0$ prevent in most real-world models of quantities?

Answer: Negative amounts that are not meaningful in context. Ensures realistic positive quantities only.

Flashcard 40: Identify the system for "$x$ and $y$ are counts, and $x+y\le 10$."

Answer: $x \ge 0,\ y \ge 0,\ x+y \le 10$. Standard constraints for counting problems.

Flashcard 41: Identify whether the point $(2,3)$ is viable for the system $x\ge 0$, $y\ge 0$, $x+y\le 4$.

Answer: Nonviable, because $2+3>4$. Sum exceeds the constraint limit.

Flashcard 42: What inequality models "at least $40$ grams of protein" if foods give $6x$ and $9y$ grams?

Answer: $6x + 9y \ge 40$. Nutritional requirement must be satisfied.

Flashcard 43: What does the constraint "at least $50$" translate to using an inequality?

Answer: $\ge 50$. Minimum required value or more.

Flashcard 44: Which phrase translates to the inequality $x \le 12$ in context?

Answer: "At most $12$.". Maximum value limitation phrase.

Flashcard 45: What does the constraint "between $3$ and $7$, not inclusive" translate to?

Answer: $3 < x < 7$. Range excludes both endpoint values.

Flashcard 46: What does the inequality symbol $\ge$ mean in a constraint?

Answer: $\ge$ means "greater than or equal to.". Allows equality at the boundary value.

Flashcard 47: What equation models "total items is $30$" if you buy $x$ of one item and $y$ of another?

Answer: $x + y = 30$. Combined quantities equal the fixed total.

Flashcard 48: Find the correct constraint for "$x$ is at least $5$ more than $y$."

Answer: $x \ge y + 5$. One variable exceeds another by minimum amount.

Flashcard 49: What inequalities represent "$x$ and $y$ are counts of items" in most models?

Answer: $x \ge 0$ and $y \ge 0$. Physical quantities cannot be negative.

Flashcard 50: What inequality models "total items is at least $30$" using $x$ and $y$?

Answer: $x + y \ge 30$. Combined quantities meet the minimum requirement.

Flashcard 51: Identify the inequality for "You must make at least $20$ units total" with $x$ and $y$ units.

Answer: $x + y \ge 20$. Minimum production requirement constraint.

Flashcard 52: What does the inequality symbol $>$ mean in a constraint?

Answer: $>$ means "strictly greater than.". Excludes the boundary value itself.

Flashcard 53: Identify the correct inequality for "spend at least $120" with costs $5x$ and $8y$.

Answer: $5x + 8y \ge 120$. Must spend at least the minimum amount.

Flashcard 54: Identify the constraint for "revenue minus cost is nonnegative" if revenue is $R$ and cost is $C$.

Answer: $R - C \ge 0$. Profitable operation requirement.

Flashcard 55: What does the inequality symbol $>$ mean in a constraint?

Answer: $>$ means "strictly greater than.". Excludes the boundary value itself.

Flashcard 56: Identify whether $(3,1)$ satisfies the constraint $2x + y \le 7$.

Answer: Viable, because $2(3)+1=7$. Satisfies the constraint exactly at boundary.

Flashcard 57: What inequality represents "profit is at least $0" if profit is $P$?

Answer: $P \ge 0$. Break-even or better financial performance.

Flashcard 58: What does it mean for a solution to be "nonviable" in a modeling context?

Answer: It violates at least one constraint or is unrealistic in context. Either mathematically invalid or contextually unrealistic.

Flashcard 59: Identify the constraint for "$x$ is no more than twice $y$."

Answer: $x \le 2y$. Sets upper bound as double the other variable.

Flashcard 60: What does the inequality symbol $\le$ mean in a constraint?

Answer: $\le$ means "less than or equal to.". Allows equality at the boundary value.

Flashcard 61: Which phrase translates to the inequality $x \ge 12$ in context?

Answer: "At least $12$.". Minimum value requirement phrase.

Flashcard 62: Identify whether the point $(1,3)$ is viable for the system $x\ge 0$, $y\ge 0$, $x+y\le 4$.

Answer: Viable, because $1+3=4$ and both are nonnegative. Satisfies all system requirements exactly.