Exercises: Linear inequalities in two variables: systems

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.

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 $D(4, 0)$.

Graphically represent the system formed by the inequalities $x + 2y \ge 4$ and $x - y \le 1$, clearly shading the resulting feasible region.

You can do it on paper and send a photo with your answer.

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.

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 $x + y \le 4$.

Represent the region of the plane that simultaneously satisfies the following constraints: $3x + y \ge 6$, $x \le 4$ and $y \le 3$.

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 $3x + 2y \ge 1$.

Represent the solution set of the system formed by the inequalities $x + y \ge 2$ and $x + y \le 5$. Describe what type of geometric region is obtained.

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 $1\,200$ pesos per hectare and barley seed costs $1\,800$ pesos per hectare. The maximum budget for seeds is $72\,000$ pesos. 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.

The figure shows a polygonal region in the Cartesian plane representing the viable production combinations of two electronic components. Determine the system of inequalities that defines this region, given that its vertices are $A(1, 1)$, $B(5, 1)$, $C(4, 4)$ and $D(2, 3)$.

To transport $60$ metric tons of goods, a logistics company has large lorries with a capacity of $8$ metric tons and small lorries with a capacity of $5$ metric tons. Due to staffing constraints, no more than $10$ vehicles in total can be sent. Set up the system of inequalities that determines the number of lorries of each type that can be used and find analytically the vertices of the solution region.

Graphically represent the system of inequalities $\frac{x}{2} + \frac{y}{3} \le 1$, $x - \frac{y}{2} \ge -1$ and $y \ge 0$. Calculate the exact coordinates of the vertices of the resulting region.

You can do it on paper and send a photo with your answer.

A personal trainer plans a routine with strength exercises ($x$) and cardio exercises ($y$). The total workout time must be between $40$ and $60$ minutes. Each strength exercise lasts $5$ minutes and each cardio exercise lasts $10$ minutes. In addition, the number of strength exercises must be at least equal to the number of cardio exercises. Set up the system of inequalities and represent it graphically.

A vehicle rental company has cars and vans. The daily maintenance of a car costs $600$ pesos and that of a van costs $1\,200$ pesos. The daily maintenance budget does not exceed $18\,000$ pesos. The parking lot has a maximum capacity for $20$ vehicles. Determine the feasible region and find the maximum number of vans that can be maintained if there are exactly $5$ cars.

Calculate the area of the region of the plane defined by the following system of inequalities: $y \ge x + 2$, $y \ge -x + 2$, and $y \le 4$.

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.

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.

To organise a concert, VIP tickets ($x$) and standard tickets ($y$) are sold. The maximum capacity of the venue is $5000$ people. For safety and profitability reasons, the number of standard tickets cannot exceed three times the number of VIP tickets. Furthermore, the organisers require that at least $1000$ VIP tickets are sold. Write the system of inequalities, find the vertices of the feasible region and determine algebraically whether it is possible to sell exactly $1500$ VIP tickets and $4000$ standard tickets.

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?

A nutritionist designs a diet based on two foods, X and Y. Each portion of food X provides $20$ g of protein, $10$ g of fat and $40$ g of carbohydrates. Each portion of Y provides $15$ g of protein, $20$ g of fat and $30$ g of carbohydrates. The diet must provide at least $120$ g of protein, no more than $100$ g of fat and exactly $240$ g of carbohydrates. Set up the associated system of inequalities and equations. Is it a two-dimensional region or a line segment? Justify your answer and find the endpoints of the solution set.

The image shows a quadrilateral feasible region modelling the constraints of a software engineering project (programming hours on the $X$-axis vs testing hours on the $Y$-axis). The vertices are $P(0, 4)$, $Q(2, 5)$, $R(5, 2)$ and $S(3, 0)$. Determine the system of inequalities that defines this region exactly.

A furniture factory produces chairs and tables. Each chair requires $2$ hours of cutting and $1$ hour of assembly. Each table requires $1$ hour of cutting and $3$ hours of assembly. The workers can dedicate a maximum of $40$ hours per week to cutting and $60$ hours to assembly. In addition, under a previous contract, at least $5$ tables must be manufactured weekly, and the number of chairs cannot be less than the number of tables. Set up the system of inequalities, plot it and find all the vertices of the feasible region.

Given the system of inequalities $2x + 3y \le 12$, $x - y \ge -1$ and $y \ge 0$, analyse how the area of the feasible region changes if the first constraint is modified to $2x + 3y \le 12 + k$, with $k > 0$. Express the area of the new region in terms of $k$, assuming that the geometric shape of the resulting polygon remains a triangle.