Linear Inequalities and Optimization Practice Test

Vocabulary and concept flashcards covering linear inequalities, boundary line graphing rules, points of intersection, and constraint formulation from word problems.

Solid Boundary Line

A boundary line drawn on a coordinate graph when a linear inequality contains a non-strict inequality symbol, such as less than or equal to ($\le$) or greater than or equal to ($\ge$), indicating that points on the line are included in the solution set.

Dashed Boundary Line

A boundary line drawn on a coordinate graph when a linear inequality contains a strict inequality symbol, such as less than ($<$) or greater than ($>$), indicating that points on the line are excluded from the solution set.

Graphing the Solution Set of $y \le -3x + 2$

The graphical representation formed by drawing a solid boundary line along $y = -3x + 2$ and shading the region below the line.

Point of Intersection (System of Boundary Lines)

The coordinate point where two boundary lines meet; for the system containing boundary lines $y - 3x = 12$ and $x = -4$, substituting $x = -4$ yields $y = 3(-4) + 12 = 0$, resulting in the point $(-4, 0)$.

Boundary Lines for $\{y - 3x > 12,\, x \le -4\}$

A dashed boundary line along $y = 3x + 12$ (due to the strict inequality symbol $>$) and a solid boundary line along $x = -4$ (due to the inclusive inequality symbol $\le$).

Capacity Constraint ($x + y \le 200$)

A linear optimization constraint representing a container or machine that holds at most $200$ total units, where $x$ is the number of cans of pop and $y$ is the number of cans of juice.

Relative Comparison Constraint ($d < p$)

An inequality representing the condition that Audrey notices more people than dogs in a park, where $d$ is the number of dogs and $p$ is the number of people.

Minimum Quantity Constraint ($d \ge 12$)

An inequality representing the condition that there are at least $12$ dogs in the dog park, where $d$ represents the number of dogs.

Total Sum Constraint ($d + p \le 40$)

An inequality representing the condition that there are no more than $40$ total people and dogs combined in the dog park, where $d$ is the number of dogs and $p$ is the number of people.