Exercises: Linear inequalities with two unknowns: systems

Graphically represent the inequality $2x - 3y \le 6$. Then, indicate two points whose coordinates belong to the solution region and two points that do not belong.

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.

From the shaded region in the figure, determine the linear inequality with two unknowns that defines it. Bear in mind 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 algebraically if 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 cakes, 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 productions.

A farmer has $50$ acres to plant wheat and barley. Wheat seed costs $\$20$ per acre and barley seed costs $\$30$ per acre. The maximum budget for seeds is $\$1200$. In addition, 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 on the Cartesian plane that represents the viable production combinations of two electronic components. Determine the system of inequalities that defines this region, knowing that its vertices are $A(1, 1)$, $B(5, 1)$, $C(4, 4)$ and $D(2, 3)$.

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

A personal trainer plans a routine with strength exercises ($x$) and cardio exercises ($y$). The total training 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 $\$10$ and that of a van costs $\$20$. The daily budget for maintenance does not exceed $\$300$. 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: $y \le x + 2$, $y \ge -x + 2$ and $y \le 4$.

A refinery produces two types of alloys, A and B. Alloy A requires $3$ pounds of copper, $1$ pounds of zinc and $2$ pounds of tin per unit. Alloy B requires $1$ pounds of copper, $2$ pounds of zinc and $3$ pounds of tin. A maximum of $300$ pounds of copper, $200$ pounds of zinc and $240$ pounds of tin is available in the warehouse. Set up the system of inequalities that describes the possible productions 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 an unbounded region and determine the coordinates of the only vertex that this region has.

For the organization of 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. In addition, the organizers require that at least $1000$ VIP tickets are sold. Write the system of inequalities, find the vertices of the feasible region and determine algebraically if it is possible to sell exactly $1500$ VIP tickets and $4000$ standard tickets.

The system of inequalities $x + y \le 6$, $2x + y \le 8$, $x \ge 0$ and $y \ge k$ is given. Discuss, in terms of the parameter $k$, the number of vertices that the feasible region has. For what values of $k$ is the feasible region reduced to the empty set?

A nutritionist designs a diet based on two foods, X and Y. Each portion of food X provides $20$ oz of protein, $10$ oz of fat and $40$ oz of carbohydrates. Each portion of Y provides $15$ oz of protein, $20$ oz of fat and $30$ oz of carbohydrates. The diet must provide at least $120$ oz of protein, no more than $100$ oz of fat and exactly $240$ oz 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 feasible region in the shape of a quadrilateral, which models the limitations 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 exactly defines this region.

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, due to a previous contract, at least $5$ tables must be manufactured per week and the number of chairs cannot be less than the number of tables. Set up the system of inequalities, represent it and find all the vertices of the feasible region.

Given the system of inequalities $2x + 3y \le 12$, $x - y \ge -1$ and $y \ge 0$. Analyze 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.