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.

Lucía has to solve the inequality $2x - y < 0$. In her notebook she writes the following:

"Step 1: I draw the solid line with equation $2x - y = 0$.

Step 2: Since the point $(0, 0)$ is the easiest to substitute, I check whether it satisfies the inequality: $2\cdot 0 - 0 < 0 \implies 0 < 0$.

Step 3: Since $0 < 0$ is false, I shade the half-plane that does not contain the point $(0, 0)$".

Identify the two conceptual errors made by Lucía, explain why her reasoning is incorrect, and describe how to correct the exercise by choosing an appropriate test point.

To determine which half-plane corresponds to the solution of the inequality $y > -3x + 6$, two classmates discuss which method to use:

• Carlos states: "Since the variable $y$ is already isolated with a positive coefficient and the sign is $>$, the solution half-plane must necessarily consist of all the points located above the line, so there is no need to check any point".

• Alba replies: "It is safer to always use the point $(0, 0)$, substitute it into the original inequality, and shade the corresponding side depending on whether it is true or false".

1. a) Apply both methods to this inequality and check whether they lead to the same region.
2. b) Analyze what would happen with Carlos's direct method if the starting inequality were $-2y > 6x - 12$. What mandatory mathematical precaution does his method require compared to the apparent safety of the test point method?

To organise a party, soft drink bottles are bought at $2$ € each and bags of snacks at $3$ € each. If the maximum budget is $30$ € 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 municipal sports center offers monthly passes that combine swimming sessions ($x$, at €$4$ each) and strength training sessions ($y$, at €$6$ each). The pass requires not exceeding a monthly budget of €$48$ ($4x + 6y \le 48$), attending at least $3$ swimming sessions ($x \ge 3$), and doing at least $2$ strength sessions ($y \ge 2$).

1. a) Before calculating anything, intuitively estimate whether a user could purchase $7$ swimming sessions and $4$ strength sessions with this pass. Justify your prediction.
2. b) Check algebraically whether the estimated combination $(7, 4)$ belongs to the feasible region.
3. c) Analytically calculate the coordinates of the three vertices of the polygon that bounds the feasible region and determine the maximum number of total sessions (swimming plus strength) that this pass allows.

A region of the plane forms a closed trapezoid whose vertices are the points $A(1, 1)$, $B(5, 1)$, $C(4, 4)$, and $D(1, 4)$, with both its four sides and its entire interior area included.

1. a) Obtain the equation of each of the four lines that form the sides of the trapezoid.
2. b) Write the system of linear inequalities with two variables whose graphical solution is exactly this region.
3. c) Use a clear interior point (for example, the point $(2, 2)$) to analytically justify why you selected the direction of the inequality sign in each of the four inequalities.

A transport company has type A lorries (capacity of $3$ tonnes) and type B lorries (capacity of $5$ tonnes). They need to transport at least $45$ tonnes 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 metre of tomatoes he needs $4$ litres of water per week, and for each square metre of peppers he needs $3$ litres. He has a tank that provides a maximum of $1800$ litres 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:

$y \le 2x + 2$

$y \le -x + 5$

$y \ge 1$

An olive-growing cooperative in Jaén sells two formats of extra virgin olive oil: $5$-liter family jugs ($x$) and $1$-liter gourmet bottles ($y$). For the Christmas campaign, the following operational constraints are established:

• The total volume of oil allocated to these two formats cannot exceed $1\,500$ liters.

• Due to glass supply limitations, there are only enough materials to bottle at most $600$ gourmet bottles.

• To ensure profitability, the total volume bottled in jugs must be at least three times the total volume bottled in gourmet bottles.

• It is required to prepare at least one jug ($x \ge 1$) and at least one bottle ($y \ge 1$).

1. a) Model the problem using a system of linear inequalities with two variables, paying special attention to how to algebraically express the condition comparing the bottled volumes.
2. b) Analytically determine the vertices of the feasible region and state, with reasoning, whether it is a bounded or unbounded region.
3. c) Check whether the cooperative can fully fulfill an order of $220$ jugs and $350$ gourmet bottles while respecting all the criteria above.

The upper skylight of a greenhouse is designed on a Cartesian plane (with measurements in meters) as a glazed region bounded between a parabolic curve and a straight beam:

1. The upper boundary is a parabola with vertex at $(0, 4)$ that intersects the horizontal axis at $(-2, 0)$ and $(2, 0)$. The frame of this parabola is part of the glazed area.

2. The lower boundary is a line connecting the points $(-2, 0)$ and $(1, 3)$, but due to structural expansion, this dividing line does not include glass (dashed boundary).

3. The transparent area lies strictly above that line and below or on the parabola.

1. a) Algebraically deduce the equation of the parabola $y = ax^2 + c$ and the explicit equation of the boundary line.
2. b) Write the system of inequalities that defines the glazed area, precisely distinguishing between strict and non-strict inequality symbols.
3. c) Analytically find the two intersection points where both boundaries meet and formally reason whether or not the point $(0, 2)$ belongs to the glazed area.

Unit Challenge. Become an energy auditor for your home!

Phase 1. Real data: Take your home's latest electricity bill (or check the average kWh price on the REE website). Note the fixed cost for contracted power, the price per kWh, and the total expenditure.

Phase 2. Modelling:

1. Set up and solve a linear system to calculate the exact consumption during peak and off-peak hours, knowing the energy consumption charge (total expenditure minus fixed costs) and the relationship between both consumptions.

2. Model the average unit cost using a rational equation of the form $y = \frac{k}{x} + c$ and calculate, by solving a single-variable inequality, above what consumption $x$ the average cost drops below a set threshold.

3. Set up a two-variable inequality that limits the use of two household appliances to avoid exceeding an extra expense of $25,5$ €. Plot the feasible region in GeoGebra.

Phase 3. Proposal: Solve a problem involving a quadratic equation that determines the dimensions of a rectangular solar panel whose length exceeds its width by $0,5$ m, covering a minimum surface area of $1,5$ m². Put everything together in a brief report with your GeoGebra screenshots and energy-saving conclusions.

Evaluation criteria:

• Analytical correctness when solving equations, inequalities, and systems.

• Rigor in using real data and precision of the feasible region in GeoGebra.

• Consistency and justification of the energy-saving proposal in the final report.