Modeling with Equation/InequalityConstraints - Algebra 2
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What is the correct interpretation of a corner point (vertex) of a feasible region in linear constraints?
What is the correct interpretation of a corner point (vertex) of a feasible region in linear constraints?
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An intersection point of boundary lines that is often a candidate optimum. Corner points are where constraint boundaries intersect.
An intersection point of boundary lines that is often a candidate optimum. Corner points are where constraint boundaries intersect.
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What does a constraint mean in a modeling problem?
What does a constraint mean in a modeling problem?
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A condition that limits allowable values of variables. Constraints establish boundaries for what's mathematically and practically possible.
A condition that limits allowable values of variables. Constraints establish boundaries for what's mathematically and practically possible.
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What does it mean for a solution to be viable in a constraints model?
What does it mean for a solution to be viable in a constraints model?
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It satisfies all constraints and matches the real-world context. A viable solution meets all mathematical and practical requirements.
It satisfies all constraints and matches the real-world context. A viable solution meets all mathematical and practical requirements.
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What does it mean for a solution to be nonviable in a constraints model?
What does it mean for a solution to be nonviable in a constraints model?
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It violates at least one constraint or is unrealistic in context. Nonviable solutions fail either mathematical or practical tests.
It violates at least one constraint or is unrealistic in context. Nonviable solutions fail either mathematical or practical tests.
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What is a concise way to state the solution set of a system of inequalities in context?
What is a concise way to state the solution set of a system of inequalities in context?
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The set of all viable $(x,y)$ satisfying every constraint. The solution set includes all points meeting every constraint.
The set of all viable $(x,y)$ satisfying every constraint. The solution set includes all points meeting every constraint.
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Identify the constraint for “The total weight is at most $100$ lb” if each $x$ weighs $8$ lb and each $y$ weighs $5$ lb.
Identify the constraint for “The total weight is at most $100$ lb” if each $x$ weighs $8$ lb and each $y$ weighs $5$ lb.
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$8x+5y \le 100$. Weight constraint limits total mass within capacity.
$8x+5y \le 100$. Weight constraint limits total mass within capacity.
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What is the correct interpretation of a corner point (vertex) of a feasible region in linear constraints?
What is the correct interpretation of a corner point (vertex) of a feasible region in linear constraints?
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An intersection point of boundary lines that is often a candidate optimum. Corner points are where constraint boundaries intersect.
An intersection point of boundary lines that is often a candidate optimum. Corner points are where constraint boundaries intersect.
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Identify the system for a diet: at least $20$ g fiber ($2x+5y\ge 20$) and at most $400$ calories ($50x+80y\le 400$).
Identify the system for a diet: at least $20$ g fiber ($2x+5y\ge 20$) and at most $400$ calories ($50x+80y\le 400$).
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$2x+5y \ge 20$ and $50x+80y \le 400$. Multiple constraints model complex nutritional requirements simultaneously.
$2x+5y \ge 20$ and $50x+80y \le 400$. Multiple constraints model complex nutritional requirements simultaneously.
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What inequality models “Cost $C$ must not exceed $\$500$”?
What inequality models “Cost $C$ must not exceed $\$500$”?
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$C \le 500$. Cost constraints ensure spending stays within budget.
$C \le 500$. Cost constraints ensure spending stays within budget.
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What inequality models “Revenue $R$ must be at least $\$500$”?
What inequality models “Revenue $R$ must be at least $\$500$”?
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$R \ge 500$. Revenue constraints ensure minimum income requirements.
$R \ge 500$. Revenue constraints ensure minimum income requirements.
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Convert the compound inequality $18 \le s \le 30$ into two inequalities.
Convert the compound inequality $18 \le s \le 30$ into two inequalities.
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$s \ge 18$ and $s \le 30$. Compound inequalities split into separate minimum and maximum constraints.
$s \ge 18$ and $s \le 30$. Compound inequalities split into separate minimum and maximum constraints.
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Identify the constraint for “At least $10$ total items” when the quantities are $x$ and $y$.
Identify the constraint for “At least $10$ total items” when the quantities are $x$ and $y$.
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$x+y \ge 10$. Combined quantities must meet or exceed the minimum total.
$x+y \ge 10$. Combined quantities must meet or exceed the minimum total.
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Identify the constraint for “No more than $10$ total items” when the quantities are $x$ and $y$.
Identify the constraint for “No more than $10$ total items” when the quantities are $x$ and $y$.
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$x+y \le 10$. Combined quantities cannot exceed the maximum total.
$x+y \le 10$. Combined quantities cannot exceed the maximum total.
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What does the intersection of the graphs of two inequalities represent?
What does the intersection of the graphs of two inequalities represent?
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The set of points satisfying both inequalities simultaneously. The intersection shows points satisfying multiple constraints simultaneously.
The set of points satisfying both inequalities simultaneously. The intersection shows points satisfying multiple constraints simultaneously.
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Identify the inequality for “$y$ is at least twice $x$” using variables $x$ and $y$.
Identify the inequality for “$y$ is at least twice $x$” using variables $x$ and $y$.
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$y \ge 2x$. At least twice means greater than or equal to $2x$.
$y \ge 2x$. At least twice means greater than or equal to $2x$.
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Identify the inequality for “$y$ is no more than twice $x$” using variables $x$ and $y$.
Identify the inequality for “$y$ is no more than twice $x$” using variables $x$ and $y$.
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$y \le 2x$. No more than twice means less than or equal to $2x$.
$y \le 2x$. No more than twice means less than or equal to $2x$.
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Identify the inequality for “The difference between $x$ and $y$ is at most $5$” (single inequality form).
Identify the inequality for “The difference between $x$ and $y$ is at most $5$” (single inequality form).
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$|x-y| \le 5$. Absolute value constraints capture differences in either direction.
$|x-y| \le 5$. Absolute value constraints capture differences in either direction.
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Rewrite the constraint $|x-y|\le 5$ as a system without absolute value.
Rewrite the constraint $|x-y|\le 5$ as a system without absolute value.
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$x-y \le 5$ and $x-y \ge -5$. Absolute value inequalities split into two linear constraints.
$x-y \le 5$ and $x-y \ge -5$. Absolute value inequalities split into two linear constraints.
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What does it mean if a system of constraints has no solution in context?
What does it mean if a system of constraints has no solution in context?
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The constraints are inconsistent; there are no viable options. Conflicting constraints create an empty feasible region.
The constraints are inconsistent; there are no viable options. Conflicting constraints create an empty feasible region.
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What does it mean if a system of constraints has infinitely many solutions in context?
What does it mean if a system of constraints has infinitely many solutions in context?
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There are many viable options; the feasible region has area/length. Multiple solutions form a region rather than discrete points.
There are many viable options; the feasible region has area/length. Multiple solutions form a region rather than discrete points.
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Identify the constraint for “At least $4$ units of $x$” when $x$ must be an integer count.
Identify the constraint for “At least $4$ units of $x$” when $x$ must be an integer count.
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$x \ge 4$ and $x \in \mathbb{Z}$. Integer constraints model discrete counting situations.
$x \ge 4$ and $x \in \mathbb{Z}$. Integer constraints model discrete counting situations.
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What is the meaning of a point outside the feasible region in a modeling context?
What is the meaning of a point outside the feasible region in a modeling context?
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A nonviable option because it breaks one or more constraints. Points outside violate constraints and are impractical solutions.
A nonviable option because it breaks one or more constraints. Points outside violate constraints and are impractical solutions.
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What additional constraint models that $x$ and $y$ count items and cannot be fractional?
What additional constraint models that $x$ and $y$ count items and cannot be fractional?
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$x \in \mathbb{Z}$ and $y \in \mathbb{Z}$ (often with $x,y\ge 0$). Discrete quantities require integer domain restrictions.
$x \in \mathbb{Z}$ and $y \in \mathbb{Z}$ (often with $x,y\ge 0$). Discrete quantities require integer domain restrictions.
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A class requires at least $18$ students and at most $30$. What compound inequality models this for $s$?
A class requires at least $18$ students and at most $30$. What compound inequality models this for $s$?
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$18 \le s \le 30$. Compound inequalities capture both minimum and maximum bounds.
$18 \le s \le 30$. Compound inequalities capture both minimum and maximum bounds.
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Which check determines whether $(x,y)$ is a solution to a system of inequalities?
Which check determines whether $(x,y)$ is a solution to a system of inequalities?
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Substitute into every inequality and verify all are true. A solution must satisfy every constraint in the system.
Substitute into every inequality and verify all are true. A solution must satisfy every constraint in the system.
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Find whether $(4,5)$ satisfies $2x+y<13$.
Find whether $(4,5)$ satisfies $2x+y<13$.
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Nonviable; $2(4)+5=13$ is not less than $13$. Strict inequality requires the value be less than, not equal to $13$.
Nonviable; $2(4)+5=13$ is not less than $13$. Strict inequality requires the value be less than, not equal to $13$.
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Find whether $(3,2)$ satisfies the system $x\ge 0$, $y\ge 0$, and $x+y\le 4$.
Find whether $(3,2)$ satisfies the system $x\ge 0$, $y\ge 0$, and $x+y\le 4$.
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Nonviable; $3+2=5$ violates $x+y\le 4$. The sum $3+2=5$ exceeds the maximum allowed total of $4$.
Nonviable; $3+2=5$ violates $x+y\le 4$. The sum $3+2=5$ exceeds the maximum allowed total of $4$.
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Find whether $(1,3)$ satisfies the system $x\ge 0$, $y\ge 0$, and $x+y\le 4$.
Find whether $(1,3)$ satisfies the system $x\ge 0$, $y\ge 0$, and $x+y\le 4$.
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Viable; $1+3=4$ and both are nonnegative. All constraints are satisfied: $1 \ge 0$, $3 \ge 0$, and $4 \le 4$.
Viable; $1+3=4$ and both are nonnegative. All constraints are satisfied: $1 \ge 0$, $3 \ge 0$, and $4 \le 4$.
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What is the meaning of the boundary line being included in the solution set?
What is the meaning of the boundary line being included in the solution set?
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The inequality uses $\le$ or $\ge$, so equality is allowed. Non-strict inequalities include the boundary as valid solutions.
The inequality uses $\le$ or $\ge$, so equality is allowed. Non-strict inequalities include the boundary as valid solutions.
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What inequality symbol models “at least” in a constraint statement?
What inequality symbol models “at least” in a constraint statement?
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$\ge$. Greater than or equal to includes the minimum value.
$\ge$. Greater than or equal to includes the minimum value.
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