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.

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.

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 1st Baccalaureate student wants to organise a graduation party. The cost of hiring the venue is $200$ € and each meal costs $15$ €. If the total available budget is at most $1500$ € 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.

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.

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.

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.

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.