Exercises: Inequalities with two unknowns

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

• $A(0, 0)$

• $B(4, 1)$

• $C(2, -1)$

• $D(-2, -5)$

Write the linear inequality in two unknowns whose solution region is represented in the following graph. Pay attention to whether the boundary line is solid or dashed.

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

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

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 in two unknowns: "Three times the weight of a large box ($x$) plus twice the weight of a small box ($y$) does not exceed $50$ kg".

Graphically represent the system of inequalities that defines the fourth quadrant of the Cartesian plane (where the x-coordinate is non-negative and the y-coordinate is non-positive).

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

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

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

To organise a party, soft drink bottles are bought at $40$ pesos each and bags of snacks at $60$ pesos each. If the maximum budget is $600$ pesos and at least one unit of each product must be bought, 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:

Graphically represent the feasible region defined by the following system of inequalities:

$2x + y \le 6$

$x - y \ge -1$

$y \ge 0$

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

In a test with $20$ questions, each correct answer adds $2$ points and each incorrect answer subtracts $1$ point. If a candidate needs at least $15$ points to pass, set up the inequality relating the number of correct answers ($x$) and the number of incorrect answers ($y$). Given that $x \ge 0$, $y \ge 0$ and $x+y \le 20$, graphically represent the region of possible combinations to pass.

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

Represent 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 the number 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 $kx - 2y \le 3$.

Graphically represent the solution to the quadratic inequality in two unknowns $y \ge x^2 - 4x + 3$.

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

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

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

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

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

$x - y \le 1$

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

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

Given the system of inequalities:

$x + 2y \le 7$

$ax - y \ge 1$

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

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

Write a system of inequalities whose solution is a square centred at the origin, with a side length of $4$ units, with its sides parallel to the Cartesian axes, and where the boundary points are included.

Solve the following system graphically and find analytically the coordinates of the vertices of the feasible region:

$y \le 2x + 2$

$y \le -x + 5$

$y \ge 1$