Inequalities and Modelling Questionbank

Two boundary lines in a system of linear inequalities are parallel. If the shaded regions are directed away from each other such that they do not overlap, the system has

A feasible region is defined by the following constraints: $x \ge 0$, $y \ge 0$, $x + 2y \le 8$, and $x + y \le 6$. Find the coordinates of the vertex that is the intersection of the two non-axis boundary lines.

Solve the inequality for $x$:

A bakery makes $x$ loaves of bread and $y$ cakes. Each loaf needs 3 cups of flour and each cake needs 2 cups. The bakery has a maximum of 60 cups of flour available. Which inequality represents this constraint?

Solve the double inequality for $x$:

Express the result in interval notation.

![ A coordinate plane showing a dashed line that intersects the y-axis at 3 and the x-axis at 4. The region below and to the left of the line, including the origin (0,0), is shaded. ] Identify the inequality representing the shaded region.

A system of two linear inequalities in two variables has no solution if the two shaded regions

Consider a system of two inequalities in two variables: $x + y > 5$ and $x + y < 2$. Which of the following describes the solution set?

In a linear programming maximization problem with the objective function $Z = 12x + 15y$ and constraints $x \geq 0, y \geq 0, x + y \leq 10$, what is the value of the objective function at the vertex representing no production ($x=0, y=0$)?

A linear programming problem requires minimizing the objective function $C = 3x + 2y$. The vertices of the feasible region are $(2, 8)$ and $(7, 3)$. Determine the minimum value of $C$.