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 $200$ € 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.

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.

To graphically solve the system of inequalities:

$$\begin{cases} 3x - 2y \geq 0 \\ x + y \leq 6 \end{cases}$$

Cian follows this procedure:

1. He graphs the boundary lines $r_1: 3x - 2y = 0$ and $r_2: x + y = 6$ on the plane.

2. To decide which half-plane to choose for the first inequality, he takes the origin $(0,0)$ as a test point, checks that $3\times 0 - 2\times 0 \geq 0$ is true ($0 \geq 0$), and concludes that he must shade the half-plane containing $(0,0)$.

3. For the second inequality, he takes $(0,0)$ again, verifies that $0 + 0 \leq 6$ is true, and shades the half-plane containing the origin.

4. He determines the solution as the intersection of both half-planes.

Identify and explain with reasoning the serious conceptual error Cian makes in step 2. Explain why the origin cannot be used to make that decision for that specific line, propose a suitable test point that resolves the ambiguity, and describe which is the correct half-plane for the first inequality.

Consider the following system of inequalities with one unknown:

$$\begin{cases} 5 - 2x \leq 13 \\ \frac{4x - 1}{3} < 5 \end{cases}$$

1. a) Before carrying out any algebraic calculations, make a reasoned estimate: mentally substitute the values $x = -5$ and $x = 2$ into both inequalities. Deduce, without solving for the unknown, whether either of these two values can be part of the solution set, and justify your estimate based on the magnitude of the terms.
2. b) Formally solve the system by finding the solution interval of each inequality and their final intersection.
3. c) Check whether your previous estimates were correct and explain what happens to the direction of the inequality sign in the first inequality when solving for the unknown, reasoning why this change occurs geometrically on the real number line.

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 $2000$ € per month. Each radio ad costs $200$ € and each poster campaign $100$ €. 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.

An organic olive oil mill in Cork markets two formats of extra virgin olive oil: Family Jug ($x$, 5 liters) and Early Harvest Bottle ($y$, 1 liter). For the November campaign, management sets the following production constraints:

• Due to the maximum capacity of the bottling line, the total number of containers (jugs and bottles) cannot exceed $1000$ units per week.

• To ensure profitability, the number of Early Harvest bottles bottled must be at least triple the number of Family jugs.

• Due to prior commitments with local distributors, at least $150$ Family jugs must be bottled per week.

• Due to limited availability of early fruit, a maximum of $800$ Early Harvest bottles can be produced per week.

1. a) Write the system of linear inequalities that models the situation, including the corresponding non-negativity constraints.
2. b) Algebraically determine the coordinates of all vertices of the feasible region by solving the systems of two linear equations corresponding to the intersections of the boundary lines.
3. c) Analytically justify whether it would be feasible to fulfill an extraordinary order of $200$ Family jugs and $700$ Early Harvest bottles in the same week by checking whether this point satisfies all inequalities.

On an urban planning map, a pedestrian green area has been defined in the shape of a triangle whose vertices are the points $A(1, 2)$, $B(7, 2)$, and $C(3, 6)$, with its sides included in the region.

1. a) Determine the equations of the three boundary lines that delimit the triangle: the horizontal side $AB$ and the slanted sides $AC$ and $BC$.
2. b) Construct the system of linear inequalities with two unknowns whose feasible region is exactly this closed triangle. Explain the reasoning you use to decide the direction of each inequality ($\leq$ or $\geq$), indicating which interior test point you use to confirm it.
3. c) If the town council decides to expand the area by shifting boundary $BC$ parallelly outwards so that the new vertex becomes $C'(3, 8)$ while keeping base $AB$ fixed, what would be the new inequality that replaces that of side $BC$?

A recording studio for young talents in Dublin 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 $\text{€}\ 10$ per bracelet and $\text{€}\ 30$ per necklace. The total materials budget cannot exceed $\text{€}\ 450$ 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 shop.

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?

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

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 centre 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 centre to be profitable, the total revenue must be at least $\text{€}\ 3600$. Yoga costs $\text{€}\ 30/\text{month}$ and Pilates $\text{€}\ 60/\text{month}$.

• One instructor is required for every 20 Yoga students and one for every 10 Pilates students. The centre 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).

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

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.

To study the following system of linear inequalities:

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

two students debate what the best strategy is to check whether the feasible region is bounded (a closed polygon) or unbounded (extends indefinitely):

• Sara's method: Meticulously graph the five boundary lines on graph paper using tables of values, shade the half-planes with arrows, visually look for where they intersect, and deduce the shape of the region from the resulting drawing.

• David's method: Analyze the relative position of the boundary lines (in particular, observing that the first two inequalities represent a strip delimited by parallel lines with positive slope), check whether the other constraints close this strip in the first quadrant by studying their combined behavior as the variables increase, and calculate only the essential vertices.

1. a) Critically compare both methods: discuss under what circumstances Sara's graphical method might lead to errors or require unmanageable paper dimensions, and what analytical rigor advantages David's strategy provides.
2. b) Apply the analytical reasoning proposed by David to prove whether the region is bounded or unbounded, formally justifying what happens to the points in the region when the variable $x$ becomes arbitrarily large.
3. c) Algebraically calculate the coordinates of all the vertices that delimit the feasible region.

An express courier company operating in Dublin's Low Emission Zone manages its daily fleet by combining standard electric vans ($x$) and large-capacity electric vans ($y$). The overnight recharging conditions and warehouse capacity are given by the system:

$$\begin{cases} x + y \geq 10 \\ 2x + y \leq 24 \\ x - y \leq k \\ x \geq 0, \quad y \geq 0 \end{cases}$$

where $k$ is a real parameter set by technical management according to the active environmental protocol.

1. a) Without taking into account the inequality containing the parameter $k$, analytically calculate the coordinates of the four vertices of the quadrilateral determined by the remaining four constraints in the first quadrant.
2. b) Evaluate with reasoning the values that the expression $x - y$ takes at the vertices found in the previous section, and deduce for which values of parameter $k$ the complete system is inconsistent (that is, the feasible region is completely empty).
3. c) For the specific value $k = 2$, determine the exact coordinates of all vertices of the resulting feasible region and reason what is the maximum integer number of standard vans ($x$) that the company can put into service while simultaneously satisfying all conditions.