Modeling with Equation/InequalityConstraints - Algebra 2

A storage limit allows at most $200$ liters: container $x$ is $10$ L and $y$ is $25$ L. What constraint?

$10x+25y \le 200$ . Total volume equals container sizes times quantities, within limit.
What inequality symbol models "more than" (strictly) in a constraint statement?

$>$ . Strict inequality excludes the boundary value.
What additional constraint models that $x$ and $y$ count items and cannot be fractional?

$x \in \mathbb{Z}$ and $y \in \mathbb{Z}$ (often with $x,y\ge 0$ ). Discrete quantities require integer domain restrictions.
Identify the inequality for " $y$ is at least twice $x$ " using variables $x$ and $y$ .

$y \ge 2x$ . At least twice means greater than or equal to $2x$ .
Which line style is used for a strict inequality such as $y>3x-2$ ?

Dashed boundary line. Dashed lines show the boundary is not included in solutions.
A nutrition rule requires at least $30$ g protein: food $x$ has $6$ g, food $y$ has $3$ g. What constraint?

$6x+3y \ge 30$ . Total protein equals grams per food times amounts, meeting minimum.
What inequality symbol models "at least" in a constraint statement?

$\ge$ . Greater than or equal to includes the minimum value.
What is the correct interpretation of a corner point (vertex) of a feasible region in linear constraints?

An intersection point of boundary lines that is often a candidate optimum. Corner points are where constraint boundaries intersect.
Find whether $(4,5)$ satisfies $2x+y<13$ .

Nonviable; $2(4)+5=13$ is not less than $13$ . Strict inequality requires the value be less than, not equal to $13$ .
A budget is $50 with items costing $3 and $5. What constraint uses $x$ and $y$ ?

$3x+5y \le 50$ . Total cost equals unit costs times quantities, bounded by budget.
What is the meaning of the boundary line being excluded from the solution set?

The inequality uses $<$ or $>$ , so equality is not allowed. Strict inequalities exclude the boundary from valid solutions.
What inequality models "Spend at least $200" with $x$ items at $15 and $y$ items at $25?

$15x+25y \ge 200$ . Total spending equals prices times quantities, meeting minimum.
Identify the inequality for " $y$ is no more than twice $x$ " using variables $x$ and $y$ .

$y \le 2x$ . No more than twice means less than or equal to $2x$ .
What is a concise way to state the solution set of a system of inequalities in context?

The set of all viable $(x,y)$ satisfying every constraint. The solution set includes all points meeting every constraint.
Identify the correct translation: "Fewer than $20$ students" for variable $s$ .

$s < 20$ . Fewer than means strictly less than the given number.
What does the boundary line represent when graphing an inequality like $2x+y\le 10$ ?

The equation $2x+y=10$ (where equality holds). The boundary shows where the inequality becomes an equality.
What does it mean if a system of constraints has no solution in context?

The constraints are inconsistent; there are no viable options. Conflicting constraints create an empty feasible region.
What is the standard test-point method used to decide which side of a boundary to shade?

Substitute a test point (often $(0,0)$ ) into the inequality. Test points determine which side of the boundary satisfies the inequality.
Identify the constraint for "The total weight is at most $100$ lb" if each $x$ weighs $8$ lb and each $y$ weighs $5$ lb.

$8x+5y \le 100$ . Weight constraint limits total mass within capacity.
Rewrite the constraint $|x-y|\le 5$ as a system without absolute value.

$x-y \le 5$ and $x-y \ge -5$ . Absolute value inequalities split into two linear constraints.
What is the meaning of the boundary line being included in the solution set?

The inequality uses $\le$ or $\ge$ , so equality is allowed. Non-strict inequalities include the boundary as valid solutions.
What type of mathematical statement is typically used to represent a budget limit?

An inequality such as $\text{cost} \le \text{budget}$ . Budget constraints limit total spending to available funds.
What inequality models "Cost $C$ must not exceed $500"?

$C \le 500$ . Cost constraints ensure spending stays within budget.
What inequality symbol models "at most" in a constraint statement?

$\le$ . Less than or equal to includes the maximum value.
What inequality symbol models "less than" (strictly) in a constraint statement?

$<$ . Strict inequality excludes the boundary value.
Identify the correct translation: "No more than $12$ items" for variable $x$ .

$x \le 12$ . No more than means less than or equal to the limit.
What equation models "The total number of tickets is exactly $120$ " using $a$ adult and $c$ child tickets?

$a+c=120$ . Exact totals require equality constraints, not inequalities.
What does it mean for a solution to be nonviable in a constraints model?

It violates at least one constraint or is unrealistic in context. Nonviable solutions fail either mathematical or practical tests.
Identify the constraint for "At least $4$ units of $x$ " when $x$ must be an integer count.

$x \ge 4$ and $x \in \mathbb{Z}$ . Integer constraints model discrete counting situations.
Find whether $(3,2)$ satisfies the system $x\ge 0$ , $y\ge 0$ , and $x+y\le 4$ .

Nonviable; $3+2=5$ violates $x+y\le 4$ . The sum $3+2=5$ exceeds the maximum allowed total of $4$ .
Identify the constraint for "No more than $10$ total items" when the quantities are $x$ and $y$ .

$x+y \le 10$ . Combined quantities cannot exceed the maximum total.
Find whether $(1,3)$ satisfies the system $x\ge 0$ , $y\ge 0$ , and $x+y\le 4$ .

Viable; $1+3=4$ and both are nonnegative. All constraints are satisfied: $1 \ge 0$ , $3 \ge 0$ , and $4 \le 4$ .
What does the feasible region represent for a system of inequalities?

All points that satisfy every inequality in the system. The overlap region contains all points meeting every constraint.
What does a constraint mean in a modeling problem?

A condition that limits allowable values of variables. Constraints establish boundaries for what's mathematically and practically possible.
What inequality models "Revenue $R$ must be at least $500"?

$R \ge 500$ . Revenue constraints ensure minimum income requirements.
What constraint is usually added to represent that a quantity cannot be negative?

$x \ge 0$ (nonnegativity constraint). Physical quantities like time, distance, or count cannot be negative.
Identify the inequality for "The difference between $x$ and $y$ is at most $5$ " (single inequality form).

$|x-y| \le 5$ . Absolute value constraints capture differences in either direction.
Identify the constraint for "At least $10$ total items" when the quantities are $x$ and $y$ .

$x+y \ge 10$ . Combined quantities must meet or exceed the minimum total.
Identify the correct translation: "At least $5$ hours" for variable $h$ .

$h \ge 5$ . At least means greater than or equal to the minimum.
What does the intersection of the graphs of two inequalities represent?

The set of points satisfying both inequalities simultaneously. The intersection shows points satisfying multiple constraints simultaneously.
Convert the compound inequality $18 \le s \le 30$ into two inequalities.

$s \ge 18$ and $s \le 30$ . Compound inequalities split into separate minimum and maximum constraints.
What does it mean if a system of constraints has infinitely many solutions in context?

There are many viable options; the feasible region has area/length. Multiple solutions form a region rather than discrete points.
What inequality models "Do not exceed $300$ calories" if food $x$ has $120$ calories and food $y$ has $80$ ?

$120x+80y \le 300$ . Total calories equals calories per food times amounts, staying within limit.
Labor takes $2$ hours per unit of $x$ and $1$ per unit of $y$ , with at most $40$ hours. What constraint?

$2x+y \le 40$ . Total time equals hours per unit times quantities, bounded by limit.
A class requires at least $18$ students and at most $30$ . What compound inequality models this for $s$ ?

$18 \le s \le 30$ . Compound inequalities capture both minimum and maximum bounds.
Identify the correct translation: "More than $8$ miles" for variable $m$ .

$m > 8$ . More than means strictly greater than the given value.
What does it mean for a solution to be viable in a constraints model?

It satisfies all constraints and matches the real-world context. A viable solution meets all mathematical and practical requirements.
Which phrase indicates an equality constraint rather than an inequality constraint?

"Exactly" (models with $=$ ). Exact amounts require equality rather than inequality constraints.
Which line style is used for a non-strict inequality such as $y\le 3x-2$ ?

Solid boundary line. Solid lines show the boundary is included in solutions.
Which check determines whether $(x,y)$ is a solution to a system of inequalities?

Substitute into every inequality and verify all are true. A solution must satisfy every constraint in the system.
What system models " $x$ and $y$ are nonnegative"?

$x \ge 0$ and $y \ge 0$ . Both variables must be non-negative in most real contexts.
Identify the system for a diet: at least $20$ g fiber ( $2x+5y\ge 20$ ) and at most $400$ calories ( $50x+80y\le 400$ ).

$2x+5y \ge 20$ and $50x+80y \le 400$ . Multiple constraints model complex nutritional requirements simultaneously.
What is the meaning of a point outside the feasible region in a modeling context?

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

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