6.2 Systems of Linear Inequalities in Two Variables
A system of linear inequalities consists of a combination of two or more linear inequalities involving the same variables. In real-life scenarios such as transportation planning, production, and budgeting, decisions are constrained by multiple simultaneous conditions. The solution is the set of all coordinate points \((x, y)\) that satisfy every inequality in the system simultaneously, represented graphically as an overlapping common region.
Learning Objectives
- Represent real-world situations and constraints in the form of a system of linear inequalities.
- Make and verify conjectures about points inside and outside regions that satisfy a system of linear inequalities.
- Determine, construct boundary lines for, and shade the common region that satisfies a system of linear inequalities.
- Identify the system of linear inequalities that defines a given shaded region on a Cartesian plane.
- Solve real-world contextual problems using graphical solutions of systems of linear inequalities.
Identifying Common Feasible Regions for Systems of Linear Inequalities
Overlapping shaded half-planes define the common solution set. Solid lines include boundary points; dashed lines exclude them.
The shaded region satisfies \(2y \ge x\), \(x + y < 4\) and \(x \ge 0\). The dashed line \(x + y = 4\) is excluded from the region; the solid lines \(2y = x\) and \(x = 0\) are included.
Brief Explanation
What is a System of Linear Inequalities?
A system of linear inequalities combines several individual inequalities. Instead of having an infinite single half-plane as a solution, the solution is the common region where all constraints are satisfied at the same time.
Non-negativity Constraints
When variables \(x\) and \(y\) represent quantities of physical items or people, non-negativity constraints apply: \(x \ge 0\) and \(y \ge 0\). Graphically, this restricts solutions to the first quadrant.
| Example of situation | Linear inequality |
|---|---|
| \(y\) is greater than \(x\) | \(y > x\) |
| \(y\) is less than \(x\) | \(y < x\) |
| \(y\) is not less than \(x\) | \(y \ge x\) |
| \(y\) is not more than \(x\) | \(y \le x\) |
| \(y\) is at least \(k\) times \(x\) | \(y \ge kx\) |
| \(y\) is at most \(k\) times \(x\) | \(y \le kx\) |
| Maximum of \(y\) is \(k\) | \(y \le k\) |
| Minimum of \(y\) is \(k\) | \(y \ge k\) |
| Sum of \(x\) and \(y\) is greater than \(k\) | \(x + y > k\) |
| Difference between \(y\) and \(x\) is less than \(k\) | \(y - x < k\) |
| \(y\) is more than \(x\) by at least \(k\) | \(y - x \ge k\) |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Write a linear inequality for the statement: “The number of brand \(Y\) shirts, \(y\), is at least twice the number of brand \(X\) shirts, \(x\).”
Given the system of linear inequalities \(x + y < 8\) and \(y \ge 3\), test whether the coordinate point \((4, 2)\) satisfies the system.
What type of boundary line should be drawn for the inequality \(x + y < 4\), and why?
From Region to System, and Maximum Values
Writing the system of inequalities from a shaded region, and reading a maximum value off the feasible region.
Writing the system of inequalities from a shaded region
Identify each boundary line, then ask two questions about each: which side the shaded region lies on (above/below, left/right), and whether the line is solid or dashed. The answers to those two questions give one inequality. Remember the \(x\)-axis is \(y = 0\) and the \(y\)-axis is \(x = 0\).
The shaded region is bounded by three lines: \(x + y = 8\) (dashed), the \(x\)-axis and the \(y\)-axis (both solid).
Determining a maximum value from a feasible region
The maximum number of passengers on a train in a theme park is 30. As a safety measure, the number of adults must always be greater than or equal to the number of children. Let \(x\) be the number of children and \(y\) be the number of adults.
From the graph, determine the maximum number of children allowed to board the train. If there are 18 children, can they all board the train at the same time?
| Condition | Inequality |
|---|---|
| At most 30 passengers | \(x + y \leqslant 30\) |
| Adults not fewer than children | \(y \geqslant x\) |
| Numbers cannot be negative | \(x \geqslant 0\), \(y \geqslant 0\) |
The feasible region is the triangle with vertices \((0, 0)\), \((0, 30)\) and \((15, 15)\). The vertex \((15, 15)\) is the point with the largest \(x\)-value in the region.
Common Mistakes
- Confusing “at least” with \(\le\) and “at most” with \(\ge\). In mathematics, at least means greater than or equal to (\(\ge\)), while at most means less than or equal to (\(\le\)).
- Drawing solid lines for strict inequalities (\(<\) or \(>\)), or drawing dashed lines for inclusive inequalities (\(\le\) or \(\ge\)).
- Forgetting to include the non-negativity constraints \(x \ge 0\) and \(y \ge 0\) when representing real-life quantities such as number of people or items.
- Shading individual regions separately without clearly highlighting or filling only the common intersection region.
Misconceptions
Any point lying in the shaded region of at least one inequality is a solution to the system.
A point is a solution to the system only if it simultaneously satisfies all inequalities in the system; it must lie within the intersection.
Points lying directly on any boundary line are always part of the solution.
Points lying on a boundary line are only included if that line is solid (\(\le\) or \(\ge\)). If the line is dashed (\(<\) or \(>\)), points on the line are excluded.
The inequality \(y - x \le 6\) means the difference must be strictly 6.
It defines an entire half-plane where the value of \(y - x\) can be any real number less than or equal to 6.
Summary
- A system of linear inequalities combines two or more linear inequalities representing simultaneous conditions.
- Its solution is the intersection of all satisfying regions on a Cartesian plane.
- Boundary lines are drawn solid for \(\le\) and \(\ge\), and dashed for \(<\) and \(>\).
- In real-world applications, non-negativity conditions (\(x \ge 0, y \ge 0\)) restrict solutions to the first quadrant.
- The shaded feasible region allows students to determine maximum, minimum, and allowable values.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 A factory produces \(x\) units of toy cars and \(y\) units of toy boats. The total production is at most 100 units, and the number of toy boats is not less than the number of toy cars. Which system of linear inequalities represents these conditions?
2 Which of the following points satisfies the system of inequalities \(y \le 2x\) and \(y \ge 4\)?
B. Fill in the Blanks
3 A boundary line on a Cartesian plane is drawn with a line when the linear inequality uses the symbol \(<\) or \(>\).
4 The mathematical inequality that represents the statement “the minimum value of \(y\) is 15” is .
C. Drag and Drop: Match Verbal Constraints with Linear Inequalities
Drag each verbal constraint to its matching linear inequality. On a phone, tap the answer choice first, then tap the matching box.
