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: 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 2: What inequality represents “$x$ is nonnegative” in a modeling context?

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

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

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

Flashcard 4: What does the constraint “between $3$ and $7$, inclusive” translate to?

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

Flashcard 5: What does the constraint “exactly $50$” translate to as an equation?

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

Flashcard 6: What does the constraint “no less than $50$” translate to using an inequality?

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

Flashcard 7: 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 8: Identify the constraint for “$x$ exceeds $y$ by at least $3$.”

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

Flashcard 9: 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 10: 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 11: What inequality represents “profit is at least \0$” if profit is$P$?

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

Flashcard 12: 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 13: 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 14: Which phrase translates to the inequality $x \ge 12$ in context?

Answer: “At least $12$.”. Minimum value requirement phrase.

Flashcard 15: Which phrase translates to the inequality $x \le 12$ in context?

Answer: “At most $12$.”. Maximum value limitation phrase.

Flashcard 16: 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 17: 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.

Flashcard 18: Identify whether $(4,0)$ satisfies the constraint $2x + y \le 7$.

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

Flashcard 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: Identify the constraint for “$x$ is at least twice $y$.”

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

Flashcard 26: Identify the constraint for “$x$ is no more than twice $y$.”

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

Flashcard 27: 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 28: 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 29: 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 30: 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.