Systems of Inequalities
Wilkin has $\text{RD}\$600$ to buy sodas at $\text{RD}\$60$ each and juices at $\text{RD}\$40$ each for the school party. Which combinations of sodas and juices can he buy without going over his budget? A single inequality with two unknowns can no longer be solved with an interval: its solution is a region of the plane. And if there are other conditions as well, such as the space in the cooler, each one cuts down that region. On this page you will learn to solve systems of inequalities, first with one variable and then with two.
Systems with one variable: intersection of intervals
A system of inequalities requires that all of them hold at the same time. With one unknown, you solve each inequality separately (quadratic ones too, using roots and signs by intervals) and keep the intersection of the solutions.
An inequality with two unknowns is a region
Given the system of linear inequalities $x + y < 5$ and $2x - y \ge 0$, determine which of the following points are solutions to the system: $A(2, 1)$, $B(0, 0)$,…
Given the system of linear inequalities $x + y < 5$ and $2x - y \ge 0$, determine which of the following points are solutions to the system: $A(2, 1)$, $B(0, 0)$, $C(3, 3)$ and $D(4, 0)$.
Check analytically whether the point $P(1, 2)$ belongs to the region defined by the system of inequalities: $2x + y \le 5$, $x - 3y \ge -6$ and $3x + 2y \ge 1$.
Check analytically whether the point $P(1, 2)$ belongs to the region defined by the system of inequalities: $2x + y \le 5$, $x - 3y \ge -6$ and $3x + 2y \ge 1$.
Graphing the region: boundary line and half-plane
To draw the solution:
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Draw the boundary line as if it were an equation (two points are enough).
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Make it solid if the inequality includes the equal sign and dashed if it is strict.
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Choose a test point that is not on the line and shade the side that contains it if it satisfies the inequality (and the other side if it does not).
The figure shows $2x - y < 4$: the origin satisfies $0 < 4$, so its side is shaded, the upper left of the line. The line is dashed.
Represent the inequality $2x - 3y \le 6$ graphically. Then, state two points whose coordinates belong to the solution region and two points that do not. You can do it on…
Represent the inequality $2x - 3y \le 6$ graphically. Then, state two points whose coordinates belong to the solution region and two points that do not.
You can do it on paper and send a photo with your answer.
From the shaded region in the figure, determine the linear inequality in two variables that defines it. Note that the boundary line passes through the points $(0, 2)$…
From the shaded region in the figure, determine the linear inequality in two variables that defines it. Note that the boundary line passes through the points $(0, 2)$ and $(3, 0)$ and that the region includes the origin.
Represent the region of the plane that simultaneously satisfies the following constraints: $3x + y \ge 6$, $x \le 4$ and $y \le 3$.
Represent the region of the plane that simultaneously satisfies the following constraints: $3x + y \ge 6$, $x \le 4$ and $y \le 3$.
Systems: feasible region and vertices
Graphically represent the system formed by the inequalities $x + 2y \ge 4$ and $x - y \le 1$, clearly shading the resulting feasible region. You can do it on paper and…
Graphically represent the system formed by the inequalities $x + 2y \ge 4$ and $x - y \le 1$, clearly shading the resulting feasible region.
You can do it on paper and send a photo with your answer.
Find algebraically the coordinates of the vertices of the feasible region defined by the following system of inequalities: $x \ge 0$, $y \ge 0$ and $x + y \le 4$.
Find algebraically the coordinates of the vertices of the feasible region defined by the following system of inequalities: $x \ge 0$, $y \ge 0$ and $x + y \le 4$.
Graphically represent the system of inequalities $\frac{x}{2} + \frac{y}{3} \le 1$, $x - \frac{y}{2} \ge -1$ and $y \ge 0$. Calculate the exact coordinates of the…
Graphically represent the system of inequalities $\frac{x}{2} + \frac{y}{3} \le 1$, $x - \frac{y}{2} \ge -1$ and $y \ge 0$. Calculate the exact coordinates of the vertices of the resulting region.
You can do it on paper and send a photo with your answer.
Regions bounded by a parabola
Systems can also mix a line and a parabola. The method is the same: each inequality is a region, and the solution is the common part. The points where the boundaries meet are found by solving the equation that results from setting them equal.
Problems: setting up the system from a word problem
When resources are limited (hours, pounds, budget, space), each resource gives an inequality. The steps are:
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Choose what $x$ and $y$ represent.
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Write one inequality for each constraint.
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Add $x\ge0$, $y\ge0$ if they are quantities (they cannot be negative).
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Draw the region and calculate the vertices.
A bakery makes two types of cake, T1 and T2. Cake T1 requires $2$ hours of preparation and $1$ hour of baking, while cake T2 requires $1$ hour of preparation and $2$…
A bakery makes two types of cake, T1 and T2. Cake T1 requires $2$ hours of preparation and $1$ hour of baking, while cake T2 requires $1$ hour of preparation and $2$ hours of baking. A maximum of $10$ hours of preparation and $8$ hours of baking are available per day. Set up the system of inequalities that models this situation (considering that the quantities cannot be negative) and graphically represent the region of possible daily production quantities.
You can do it on paper and send a photo with your answer.
To transport $60$ metric tons of goods, a logistics company has large lorries with a capacity of $8$ metric tons and small lorries with a capacity of $5$ metric tons.…
To transport $60$ metric tons of goods, a logistics company has large lorries with a capacity of $8$ metric tons and small lorries with a capacity of $5$ metric tons. Due to staffing constraints, no more than $10$ vehicles in total can be sent. Set up the system of inequalities that determines the number of lorries of each type that can be used and find analytically the vertices of the solution region.
A personal trainer plans a routine with strength exercises ($x$) and cardio exercises ($y$). The total workout time must be between $40$ and $60$ minutes. Each strength…
A personal trainer plans a routine with strength exercises ($x$) and cardio exercises ($y$). The total workout time must be between $40$ and $60$ minutes. Each strength exercise lasts $5$ minutes and each cardio exercise lasts $10$ minutes. In addition, the number of strength exercises must be at least equal to the number of cardio exercises. Set up the system of inequalities and represent it graphically.
Maximums and minimums on the feasible region
Often it is not enough to draw the region: you need to know the greatest or the least value of an expression such as $x+y$ or $60x+40y$ among all the possible points. When the region is a closed polygon, that maximum and that minimum are always reached at a vertex. So it is enough to calculate the vertices and evaluate the expression at each one.
Why? A linear expression changes at a steady rate as you move along a straight line, so it keeps increasing (or decreasing) until the end of the segment; the end of each segment is a vertex.
Another common situation is fixing one of the variables. Then you substitute its value into all the inequalities and get an interval for the other.
A vehicle rental company has cars and vans. The daily maintenance of a car costs $600$ pesos and that of a van costs $1\,200$ pesos. The daily maintenance budget does…
A vehicle rental company has cars and vans. The daily maintenance of a car costs $600$ pesos and that of a van costs $1\,200$ pesos. The daily maintenance budget does not exceed $18\,000$ pesos. The parking lot has a maximum capacity for $20$ vehicles. Determine the feasible region and find the maximum number of vans that can be maintained if there are exactly $5$ cars.
A farmer has $50$ hectares available to plant wheat and barley. Wheat seed costs $1\,200$ pesos per hectare and barley seed costs $1\,800$ pesos per hectare. The maximum…
A farmer has $50$ hectares available to plant wheat and barley. Wheat seed costs $1\,200$ pesos per hectare and barley seed costs $1\,800$ pesos per hectare. The maximum budget for seeds is $72\,000$ pesos. Furthermore, due to environmental regulations, the area of barley cannot exceed twice the area of wheat. Write the associated system of inequalities and find the vertices of the feasible region.
A refinery produces two types of alloys, A and B. For alloy A, $3$ kg of copper, $1$ kg of zinc and $2$ kg of tin are needed per unit. For alloy B, $1$ kg of copper, $2$…
A refinery produces two types of alloys, A and B. For alloy A, $3$ kg of copper, $1$ kg of zinc and $2$ kg of tin are needed per unit. For alloy B, $1$ kg of copper, $2$ kg of zinc and $3$ kg of tin. A maximum of $300$ kg of copper, $200$ kg of zinc and $240$ kg of tin is available in the warehouse. Set up the system of inequalities describing the possible production of both alloys, represent the feasible region and calculate the coordinates of all its vertices.
From the region to the system, areas, and special cases
From the region to the system. If you are given a polygonal region, write the equation of the line through each pair of consecutive vertices. Then, with an interior point (for example, the center of the region), decide the direction of each inequality.
Special cases:
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Two parallel lines with inequalities in opposite directions can leave a strip between them (like $2\le x+y\le5$) or an empty set (like $x+y\le2$ and $x+y\ge5$).
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If one condition is an equality (as in an "exactly" problem), the solution is a segment.
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With a parameter, like $y\ge k$, the line moves and the number of vertices changes: move the line in your mind and watch when it touches a vertex, when it cuts two sides, and when it leaves the region empty.
Bounded region? A region is bounded when it is enclosed on all sides, so that it fits inside a rectangle. Some regions are not and extend without limit in some direction. To check that a region is bounded, calculate its vertices and verify that they form a closed polygon: each vertex satisfies all the inequalities and each boundary line contributes a side between two vertices.
Represent the solution set of the system formed by the inequalities $x + y \ge 2$ and $x + y \le 5$. Describe what type of geometric region is obtained.
Represent the solution set of the system formed by the inequalities $x + y \ge 2$ and $x + y \le 5$. Describe what type of geometric region is obtained.
The figure shows a polygonal region in the Cartesian plane representing the viable production combinations of two electronic components. Determine the system of…
The figure shows a polygonal region in the Cartesian plane representing the viable production combinations of two electronic components. Determine the system of inequalities that defines this region, given that its vertices are $A(1, 1)$, $B(5, 1)$, $C(4, 4)$ and $D(2, 3)$.
Consider the region of the plane defined by the inequalities $x + 2y \ge 4$, $3x - y \le 6$, and $x - y \ge -2$. Prove algebraically that it is a bounded region and…
Consider the region of the plane defined by the inequalities $x + 2y \ge 4$, $3x - y \le 6$, and $x - y \ge -2$. Prove algebraically that it is a bounded region and determine the coordinates of the vertices of this region.
Summary
Go back to the beginning: Wilkin's region is the polygon bounded by $60x+40y\le600$ and the space in the cooler. Each point with integer coordinates inside it is a possible purchase.