Exercises: Inequalities with two unknowns

Check which of the following points are a solution to the inequality $3x - 2y \le 6$:

• $A(0, 0)$

• $B(4, 1)$

• $C(2, -1)$

• $D(-2, -5)$

Write the linear inequality with two unknowns whose solution region is represented in the following graph. Take into account whether the boundary line is solid or dashed.

Graph the solution region of the inequality $x + 2y > 4$ on the Cartesian plane.

Determine whether the points $P(1, 3)$, $Q(-1, 2)$ and $R(4, 0)$ belong to the feasible region defined by the following system of inequalities:

$x + y \le 5$

$2x - y > -1$

Translate the following statement into a linear inequality with two unknowns: "Triple the weight of a large box ($x$) plus double the weight of a small box ($y$) does not exceed $50$ lb".

Graph the system of inequalities that defines the fourth quadrant of the Cartesian plane (where the x-coordinate is positive or zero and the y-coordinate is negative or zero).

Given the inequality $4x - 5y < 20$, indicate with reasoning whether the points located exactly on the line $4x - 5y = 20$ are part of the solution region.

Write the system of inequalities whose solution is the vertical strip between $x = -2$ and $x = 3$, including both boundaries.

To organize a party, bottles of soft drink are bought at $\$2$ each and bags of snacks at $\$3$ each. If the maximum budget is $\$30$ and it is desired to buy at least one unit of each product, set up the corresponding system of inequalities and give two possible valid purchase combinations.

Find the system of inequalities that defines the shaded region in the following graph:

Graph the feasible region determined by the following system of inequalities:

$2x + y \le 6$

$x - y \ge -1$

$y \ge 0$

In a test of $20$ questions, each correct answer adds $2$ points and each mistake subtracts $1$ point. If to pass it is necessary to obtain at least $15$ points, set up the inequality that relates the number of correct answers ($x$) and the number of mistakes ($y$). Taking into account that $x \ge 0$, $y \ge 0$ and $x+y \le 20$, graph the region of possible combinations to pass.

Graph the following system of inequalities and describe in words the geometric shape of the solution region:

$3x - y > 2$

$3x - y < 6$

A workshop manufactures bicycles and tricycles. Each bicycle requires $2$ wheels and each tricycle $3$ wheels. There are $60$ wheels left in the warehouse. Furthermore, it is known that the number of bicycles manufactured must be at least double that of tricycles. Set up the system of inequalities that models this situation.

Find the values of the parameter $k$ for which the point $A(-1, 4)$ belongs to the solution region of the inequality $kx - 2y \le 3$.

Graph the solution of the quadratic inequality with two unknowns $y \ge x^2 - 4x + 3$.

A transport company has type A trucks (capacity of $3$ tons) and type B trucks (capacity of $5$ tons). They need to transport at least $45$ tons of goods. Due to staff restrictions, they cannot use more than $12$ trucks in total. Set up the system of inequalities and check analytically whether it is possible to carry out the transport using $4$ type A trucks and $7$ type B trucks.

Determine the system of inequalities that defines the bounded region shown in the following figure:

Graph the region of the plane defined by the following mixed system (a quadratic inequality and a linear inequality):

$y \le -x^2 + 2x + 3$

$x - y \le 1$

Chickens and pigs are raised on a farm. It is known that the maximum number of animals the farm can house is $100$. Furthermore, for profitability reasons, the number of pigs cannot exceed a third of the number of chickens, and there must be at least $10$ pigs. Set up the system of inequalities and graph it to find the region of possible combinations.

Given the system of inequalities:

$x + 2y \le 8$

$ax - y \ge 1$

Calculate the value of the parameter $a$ so that the point $P(3, 2)$ is a vertex of the feasible region (that is, it is located exactly at the intersection of the two boundary lines).

A farmer wants to plant tomatoes and peppers on a plot of $500 \text{ ft}^2$. For each square foot of tomatoes he needs $4$ gallons of water a week, and for each square foot of peppers he needs $3$ gallons. He has a tank that provides him with a maximum of $1800$ gallons per week. If for commercial reasons he must dedicate at least $100 \text{ ft}^2$ to tomatoes and at least $50 \text{ ft}^2$ to peppers, set up the system of inequalities and find out if he can plant $250 \text{ ft}^2$ of each vegetable.

Write a system of inequalities whose solution is a perfect square centered at the origin of coordinates, with a side of $4$ units, with its sides parallel to the Cartesian axes, and where the points on the edges are included.

Graph the following system to solve it and find analytically the coordinates of the vertices of the feasible region:

$y \le 2x + 2$

$y \le -x + 5$

$y \ge 1$