Linear inequalities and systems of linear inequalities
A catering company has to decide how many basic menus ($x$) and how many premium menus ($y$) to prepare for an event: the budget, the number of cooks available and the capacity of the hall impose several conditions on $x$ and $y$ at the same time. Each condition is a linear inequality in two unknowns; all of them together form a system. This section reviews how each piece is solved — the half-plane of an inequality, the region of a system — because on that region you will build, in the next sections, linear programming: choosing the best point from among all the possible ones.
The half-plane of a linear inequality
Represent on the plane the solution half-plane of the inequality $2x + 3y \leq 12$ and check algebraically whether the points $A(3, 2)$ and $B(2, 5)$ belong to it. What…
Represent on the plane the solution half-plane of the inequality $2x + 3y \leq 12$ and check algebraically whether the points $A(3, 2)$ and $B(2, 5)$ belong to it. What special feature does the point $A$ have with respect to the boundary line?
You can do it on paper and send a photo with your answer.
Represent the set of points in the plane satisfying $y \geq 2x - 1$. Explain how you decide which half-plane to shade and whether the line $y = 2x - 1$ belongs to the…
Represent the set of points in the plane satisfying $y \geq 2x - 1$. Explain how you decide which half-plane to shade and whether the line $y = 2x - 1$ belongs to the solution or not.
Systems of inequalities: the intersection of half-planes
Represent the solution region of the system formed by $x \geq 0$, $y \geq 0$ and $x + y \leq 5$. What geometric shape is obtained? State its vertices.
Represent the solution region of the system formed by $x \geq 0$, $y \geq 0$ and $x + y \leq 5$. What geometric shape is obtained? State its vertices.
Represent the solution region of the system formed by $x \geq 0$, $y \geq 0$, $x + y \leq 6$ and $x \leq 4$. Calculate the coordinates of all its vertices and state what…
Represent the solution region of the system formed by $x \geq 0$, $y \geq 0$, $x + y \leq 6$ and $x \leq 4$. Calculate the coordinates of all its vertices and state what type of polygon it is.
Setting up the system from a problem statement
The most important step — and the one that goes wrong most often in exams — is translating a text correctly into inequalities. Each condition in the statement ("no more than", "at least", "at most") becomes an inequality with $\leq$ or $\geq$.
A sports clothing shop is preparing its spring order: let $x$ be the number of technical t-shirts and $y$ be the number of hoodies it will order. Due to warehouse space,…
A sports clothing shop is preparing its spring order: let $x$ be the number of technical t-shirts and $y$ be the number of hoodies it will order. Due to warehouse space, it cannot receive more than $40$ garments in total, and the supplier requires it to order at least $10$ t-shirts. Write the system of inequalities that captures all these conditions.
The figure shows the solution region of a system of inequalities and four points marked on it, as shown in the figure. Decide graphically which points are solutions to…
The figure shows the solution region of a system of inequalities and four points marked on it, as shown in the figure.
Decide graphically which points are solutions to the system and confirm your answer by checking each point algebraically in the system $x + 2y \leq 10$, $3x + y \leq 15$, $x \geq 0$, $y \geq 0$.
Systems with a parameter
Some university entrance exam problems introduce a parameter into one of the constraints and ask how the region changes depending on its value. The strategy is the same: draw the region for a couple of specific values of the parameter and observe which line moves.
Consider the system formed by $x \geq 0$, $y \geq 0$, $x + y \leq 4$ and $y \leq k$, where $k > 0$ is a parameter. a) Represent the region for $k = 6$ and for $k = 2$.…
Consider the system formed by $x \geq 0$, $y \geq 0$, $x + y \leq 4$ and $y \leq k$, where $k > 0$ is a parameter.
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a) Represent the region for $k = 6$ and for $k = 2$.
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b) Determine for which values of $k$ the solution region is a quadrilateral and for which it is a triangle, justifying your answer.
Back to the start: each limit of the catering company (capacity, cooks, budget) is expressed as a linear inequality; their intersection delimits the polygonal region that contains all the feasible combinations of menus $(x,y)$.