Exercises: Systems of inequalities

Determine the set of real numbers that simultaneously satisfy the following two conditions. Express the result in interval notation and represent it on the number line:

1. Twice a number is greater than 4.

2. That same number is less than or equal to 7.

Write down the corresponding system of inequalities and find its solution.

Solve the following system of linear inequalities with a single unknown:

$\begin{cases} 2x - 5 < 3 \\ 3x + 2 \ge 8 \end{cases}$

1. a) Solve each inequality separately and express the solution of the system as an interval. Represent it on the number line.

1. b) Determine whether the value $x = 3$ is part of the solution, by checking it in both inequalities.

Represent in the Cartesian plane the region of points $(x, y)$ satisfying the linear inequality:

$x + y \le 5$

To do this, first draw the associated line, choose a test point (such as the origin) and shade the correct half-plane.

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

Identify graphically the feasible region (the region of common solutions) of the following system of inequalities in the first quadrant:

$\begin{cases} x \ge 0 \\ y \ge 0 \\ x + 2y \le 6 \end{cases}$

The figure shows the axes and the line $x + 2y = 6$, with the feasible region shaded.

1. a) Check, with the help of the figure, that the region is a triangle.

1. b) Calculate the coordinates of the three vertices that form the triangle of the solution.

A 1st Baccalaureate student wants to organise a graduation party. The cost of hiring the venue is $4\,000$ pesos and each meal costs $300$ pesos. If the total available budget is at most $30\,000$ pesos and they expect at least $40$ people to attend:

1. a) Set up a system of inequalities to model this situation, letting $x$ be the number of attendees.
2. b) Is it possible for $90$ people to attend the party? Justify your answer using the system set up.

Given the following graph representing the solution to a system of inequalities, determine which of the proposed options corresponds to the shaded constraints:

A) $\begin{cases} x \le 2 \\ y \ge 1 \end{cases}$ within the first quadrant ($x \ge 0, y \ge 0$).

B) $\begin{cases} x \ge 2 \\ y \le 1 \end{cases}$

C) $\begin{cases} x \le 2 \\ y \le 1 \end{cases}$

Check whether the points $A(1, 1)$, $B(4, 0)$ and $C(2, 3)$ are solutions to the system of inequalities:

$\begin{cases} x + y > 2 \\ 2x - y < 5 \end{cases}$

You must verify each point in both inequalities to confirm whether they belong to the region of intersection.

A small company manufactures two types of phone cases: the "Eco" model ($x$) and the "Premium" model ($y$). For production reasons, the company must satisfy these daily conditions:

1. The total number of cases manufactured cannot exceed $100$ units.

2. At least $20$ cases of the "Premium" model must be manufactured.

Set up the system of inequalities representing these constraints and sketch the region containing the possible production combinations.

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

Solve the following system of inequalities in one variable and represent the solution on the number line, expressing it in interval notation:

$\begin{cases} \frac{2x - 5}{3} \leq x + 1 \\ 3(x - 2) > 2x - 7 \end{cases}$

Determine graphically the feasible region of the plane that satisfies the following system of linear inequalities:

$$\begin{cases} 2x + y \geq 4 \\ x - 2y < 2 \\ y \leq 3 \end{cases}$$

Draw a set of coordinate axes in your notebook and solve the following parts:

1. a) Plot each of the three boundary lines. Use a solid line if the inequality is not strict and a dashed line if it is.

1. b) Choose a test point for each line, decide which half-plane corresponds to each inequality, and shade the feasible region.

1. c) State whether the point $P(1, 2)$ is part of the solution of the system. Justify it by substituting its coordinates into the three inequalities.

A technology company launches two new models of wireless headphones: the Sonic model ($x$) and the Bass model ($y$). Due to production and market constraints, the following conditions must be met:

1. The total number of units manufactured cannot exceed 80 per day.

2. To maintain exclusivity, at least 20 units of the Sonic model must be manufactured.

3. The number of units of the Bass model cannot exceed three times the Sonic model.

Write down the system of inequalities representing these constraints and draw the region of possible solutions.

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

Find the system of linear inequalities defining the shaded region in the following figure:

A nutritionist is designing a diet combining two foods, $A$ and $B$. Each unit of food $A$ provides 2 grams of protein and 4 of carbohydrates. Each unit of food $B$ provides 3 grams of protein and 2 of carbohydrates. The diet must satisfy the following minimum daily requirements:

• At least 12 grams of protein.

• At least 10 grams of carbohydrates.

• No more than 5 units of food $B$ can be consumed.

Set up the system of inequalities and represent graphically the region where all requirements are satisfied (feasible region).

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

Solve the following system of four inequalities graphically and state whether the resulting region is bounded (closed) or unbounded (open). Calculate the coordinates of all its vertices:

$\begin{cases} x + y \leq 6 \\ 2x - y \leq 4 \\ x \geq 0 \\ y \geq 0 \end{cases}$

Find the set of values of $x$ that simultaneously satisfy the following three conditions:

1. Twice $x$ increased by 10 units is greater than three times $x$ decreased by 5.

2. A quarter of $x$ is less than or equal to 4.

3. $x$ must be a positive number.

Express the solution using a single modulus inequality if possible, or as an interval.

A youth association is promoting its annual party with ads on the local radio ($x$ = number of radio ads) and posters in the secondary schools of the region ($y$ = number of poster campaigns). They have a maximum budget of $2000$ pesos per month. Each radio ad costs $200$ pesos and each poster campaign $100$ pesos. For outreach strategy reasons, they have decided that:

• The number of poster campaigns must be at least equal to the number of radio ads.

• At most 8 actions in total must be carried out across both media.

• No more than 6 poster campaigns can be carried out.

In addition, $x$ and $y$ are non-negative integers.

1. a) Write the system of inequalities that describes these conditions.

1. b) Graphically represent the decision area (feasible region).

1. c) Propose three integer combinations $(x, y)$ of actions that are valid and check that they meet all the conditions.

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

A recording studio for young talents in Madrid offers two types of sessions: Acoustic Session ($x$) and Band Session ($y$). Due to noise regulations and the availability of the technicians, the studio must meet the following weekly restrictions:

• The total number of sessions cannot exceed 15.

• The editing time for an acoustic session is 2 hours and for a band session it is 5 hours. The studio has no more than 50 editing hours per week.

• Due to promotion policies, the number of band sessions must be at least half the number of acoustic sessions.

• No more than 8 band sessions can be held.

• Also, $x \geq 0$ and $y \geq 0$, because there cannot be a negative number of sessions.

1. a) Graph the feasible region where the possible combinations of sessions lie.

1. b) Determine whether it is possible to hold 10 acoustic sessions and 6 band sessions while meeting all the requirements.

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

Determine the system of linear inequalities that defines the convex quadrilateral whose vertices are the points $A(1, 2)$, $B(5, 2)$, $C(4, 6)$ and $D(2, 5)$. You must express each inequality in the form $ax + by \leq c$ or $ax + by \geq c$.

A craft workshop produces bracelets ($x$) and necklaces ($y$) to sell at a weekly market. The following conditions for daily production are known:

1. The material cost is $\$200$ per bracelet and $\$600$ per necklace. The total materials budget cannot exceed $\$9\,000$ per day.

2. For every necklace made, at least 2 bracelets must be produced, as they are sold as a set.

3. Due to display space, the number of bracelets cannot exceed the number of necklaces by more than 10.

4. The sum of three times the number of necklaces and the number of bracelets must be at least 15 to cover the minimum order of a local store.

Set up the system of inequalities, draw the solution region and answer: what is the maximum number of necklaces that can be produced under these conditions?

Consider the following system of inequalities depending on a parameter $k \in \mathbb{R}$:

$$\begin{cases} x + y \leq 6 \\ x \geq 1 \\ y \geq 0 \\ y \leq k \end{cases}$$

Analyze how the shape of the feasible region changes (whether it is a triangle, a trapezoid, or if it disappears) depending on the values of $k$. In particular, determine the value of $k$ for which the feasible region is a trapezoid with an area of exactly $8$ square units.

Solve the following system of inequalities graphically and check whether there is an integer solution $(x, y)$ satisfying all conditions simultaneously:

$$\begin{cases} \frac{x-y}{2} + \frac{x+y}{3} > 1 \\ 2x - y < 4 \\ x + 3y > 6 \\ y < 3 \end{cases}$$

Indicate with reasons whether the feasible region is bounded or unbounded.

A sports center offers Yoga and Pilates classes. Due to health and safety regulations, the total capacity is limited to 100 people. Furthermore:

• The number of Yoga students ($x$) cannot exceed twice the number of Pilates students ($y$).

• For the center to be profitable, the total revenue must be at least $\$72\,000$. Yoga costs $\$600/\text{month}$ and Pilates $\$1\,200/\text{month}$.

• One instructor is required for every 20 Yoga students and one for every 10 Pilates students. The center has only 7 instructors available during that time slot.

Write down the simplified system of inequalities, draw the feasible region and determine the extreme points (vertices).

Consider the region of the plane defined by the following system:

$$\begin{cases} y - x \leq 4 \\ y + x \geq 2 \\ y - 2x \geq -4 \\ y \leq 5 \end{cases}$$

1. a) Draw the feasible region accurately.

1. b) Calculate the area of the resulting polygon.

1. c) Determine the points of the region that minimize the expression $Z = x + y$ (state whether there is only one or more than one), justifying your answer from the graphical representation.

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