Systems of linear inequalities in two unknowns
Until now, the solution of an inequality was an interval on the number line. With two unknowns ($x$ and $y$), the solution no longer fits on a line: it is a region of the plane. A system of several inequalities cuts that region down until, often, a polygon is left: the feasible region, the basis of the linear programming you will use in optimisation problems.
An inequality in two unknowns is a region
Given the system of linear inequalities $x + y < 5$ and $2x - y \ge 0$, determine which of the following points are solutions to the system: $A(2, 1)$, $B(0, 0)$,…
Given the system of linear inequalities $x + y < 5$ and $2x - y \ge 0$, determine which of the following points are solutions to the system: $A(2, 1)$, $B(0, 0)$, $C(3, 3)$ and $D(4, 0)$.
Represent the inequality $2x - 3y \le 6$ graphically. Then, state two points whose coordinates belong to the solution region and two points that do not. You can do it on…
Represent the inequality $2x - 3y \le 6$ graphically. Then, state two points whose coordinates belong to the solution region and two points that do not.
You can do it on paper and send a photo with your answer.
Representing the region: the boundary line and the half-plane
To draw the region, you first plot the boundary line (as if the inequality were an equation) and then decide, using a test point, which side to shade. The line is drawn dashed if the inequality is strict ($<$ or $>$) and solid if it includes the equals sign.
From the shaded region in the figure, determine the linear inequality in two variables that defines it. Note that the boundary line passes through the points $(0, 2)$…
From the shaded region in the figure, determine the linear inequality in two variables that defines it. Note that the boundary line passes through the points $(0, 2)$ and $(3, 0)$ and that the region includes the origin.
Represent the region of the plane that simultaneously satisfies the following constraints: $3x + y \ge 6$, $x \le 4$ and $y \le 3$.
Represent the region of the plane that simultaneously satisfies the following constraints: $3x + y \ge 6$, $x \le 4$ and $y \le 3$.
Systems: the feasible region and its vertices
Find algebraically the coordinates of the vertices of the feasible region defined by the following system of inequalities: $x \ge 0$, $y \ge 0$ and $x + y \le 4$.
Find algebraically the coordinates of the vertices of the feasible region defined by the following system of inequalities: $x \ge 0$, $y \ge 0$ and $x + y \le 4$.
Check analytically whether the point $P(1, 2)$ belongs to the region defined by the system of inequalities: $2x + y \le 5$, $x - 3y \ge -6$ and $3x + 2y \ge 1$.
Check analytically whether the point $P(1, 2)$ belongs to the region defined by the system of inequalities: $2x + y \le 5$, $x - 3y \ge -6$ and $3x + 2y \ge 1$.
Problems: setting up the system from a word problem
The main use of this section is to model a situation with limited resources (hours, kilograms, budget, capacity) as a system of inequalities, and to find the feasible region. It is the basis of linear programming.
A bakery makes two types of cake, T1 and T2. Cake T1 requires $2$ hours of preparation and $1$ hour of baking, while cake T2 requires $1$ hour of preparation and $2$…
A bakery makes two types of cake, T1 and T2. Cake T1 requires $2$ hours of preparation and $1$ hour of baking, while cake T2 requires $1$ hour of preparation and $2$ hours of baking. A maximum of $10$ hours of preparation and $8$ hours of baking are available per day. Set up the system of inequalities that models this situation (considering that the quantities cannot be negative) and graphically represent the region of possible daily production quantities.
You can do it on paper and send a photo with your answer.
A farmer has $50$ hectares available to plant wheat and barley. Wheat seed costs $20$ euros per hectare and barley seed costs $30$ euros per hectare. The maximum budget…
A farmer has $50$ hectares available to plant wheat and barley. Wheat seed costs $20$ euros per hectare and barley seed costs $30$ euros per hectare. The maximum budget for seeds is $1200$ euros. Furthermore, due to environmental regulations, the area of barley cannot exceed twice the area of wheat. Write the associated system of inequalities and find the vertices of the feasible region.
A refinery produces two types of alloys, A and B. For alloy A, $3$ kg of copper, $1$ kg of zinc and $2$ kg of tin are needed per unit. For alloy B, $1$ kg of copper, $2$…
A refinery produces two types of alloys, A and B. For alloy A, $3$ kg of copper, $1$ kg of zinc and $2$ kg of tin are needed per unit. For alloy B, $1$ kg of copper, $2$ kg of zinc and $3$ kg of tin. A maximum of $300$ kg of copper, $200$ kg of zinc and $240$ kg of tin is available in the warehouse. Set up the system of inequalities describing the possible production of both alloys, represent the feasible region and calculate the coordinates of all its vertices.
Unbounded regions and special cases
Consider the region of the plane defined by the inequalities $x + 2y \ge 4$, $3x - y \le 6$, and $x - y \ge -2$. Prove algebraically that it is a bounded region and…
Consider the region of the plane defined by the inequalities $x + 2y \ge 4$, $3x - y \le 6$, and $x - y \ge -2$. Prove algebraically that it is a bounded region and determine the coordinates of the vertices of this region.
Given the system of inequalities $x + y \le 6$, $2x + y \le 8$, $x \ge 0$ and $y \ge k$, discuss, in terms of the parameter $k$, the number of vertices of the feasible…
Given the system of inequalities $x + y \le 6$, $2x + y \le 8$, $x \ge 0$ and $y \ge k$, discuss, in terms of the parameter $k$, the number of vertices of the feasible region. For which values of $k$ does the feasible region reduce to the empty set?