A linear inequality in two variables, such as y > 2x – 1, has a half-plane of solutions bounded by a line. Draw the boundary dashed for < and > or solid for \(\le\) and \(\ge\), then shade the side that makes a test point true. A system’s solution is the region where all the shadings overlap.
Key facts:
- Standards A-REI.D.12 and A-CED.A.3: graph linear inequalities and systems, and model constraints.
- Solid line for \(\le\) and \(\ge\); dashed line for \(<\) and \(>\).
- Use a test point not on the line, such as \((0, 0)\), to decide which side to shade.
- Multiplying or dividing by a negative number reverses the inequality symbol.
- The solution set of a system is the overlap of the half-planes; a feasible region lists all allowed combinations.
What is a linear inequality in two variables?
A linear inequality in two variables, such as \(y>2x-1\) or \(3x+2y\le 12\), replaces the equals sign of a line with \(<\), \(\le\), \(>\) or \(\ge\). Its solutions are ordered pairs \((x, y)\) that make it true. Those points fill a whole region of the plane called a half-plane, bounded by the line you get when you change the inequality symbol to \(=\).
Boundary line: the line that separates solutions from non-solutions.
Solid line: used for \(\le\) and \(\ge\); points on the line are solutions.
Dashed line: used for \(<\) and \(>\); points on the line are not solutions.
How do you graph a linear inequality?
- Graph the boundary line, replacing the inequality symbol with an equals sign.
- Make the line dashed for \(<\) or \(>\), and solid for \(\le\) or \(\ge\).
- Pick a test point that is not on the line. The origin \((0, 0)\) is easiest.
- Substitute the test point. If the result is true, shade the side containing it. If false, shade the other side.
Worked example. Graph \(y>2x-1\). The boundary \(y=2x-1\) is dashed. Test \((0, 0)\): \(0>-1\) is true, so shade the side that contains the origin, which is above the line.
When a line is written as \(Ax+By<C\), solve for \(y\) or test a point. If you divide or multiply by a negative number, reverse the inequality symbol: \(-2y>8\) becomes \(y<-4\).
How do you graph a system of inequalities?
Graph each inequality on the same grid. The solution set of the system is the region where all the shadings overlap. Check by testing one point inside the overlap in every inequality.
Worked example. Graph \(y>x-2\) and \(y\le -x+4\). The first line is dashed and shaded above; the second is solid and shaded below. The lines meet at \((3, 1)\). That corner is an open circle because \(1>1\) is false. Test \((0, 0)\): \(0>-2\) and \(0\le 4\) are both true, so \((0, 0)\) is in the solution region.
How do constraints model real situations?
A constraint is a limit written as an inequality. Priya buys \(x\) cupcakes at \(\$2\) and \(y\) muffins at \(\$3\) with at most \(\$12\). The constraints are \(2x+3y\le 12\), \(x\ge 0\) and \(y\ge 0\). The overlap is a triangle called the feasible region. Only whole-number points make sense for items you can count.
Graphing Linear Inequalities and Systems of Inequalities: frequently asked questions
How do I know whether to use a solid or a dashed line?
Look at the symbol. Use a solid line for \(\le\) and \(\ge\), because points on the line make the statement true. Use a dashed line for \(<\) and \(>\), because points on the line do not.
Which side of the line should I shade?
Pick a test point that is not on the line, usually \((0, 0)\), and substitute it. If the inequality is true, shade the side with that point. If it is false, shade the opposite side.
Why do I flip the inequality symbol when I divide by a negative number?
Multiplying by a negative reverses the order of numbers on the number line. For example, \(2<5\) but \(-2>-5\). The same reversal keeps \(-3y>6\) true when it becomes \(y<-2\).
What is the solution set of a system of inequalities?
It is the set of all points that make every inequality true, which is the region where the shaded half-planes overlap. If the half-planes do not overlap, the system has no solution.
What is a feasible region?
A feasible region is the part of the graph where every constraint in a real-world problem is satisfied. Each point in it is a combination that follows all the limits, such as a budget and a minimum amount.
- Graphing Linear Inequalities and Systems of Inequalities worksheet (22 problems)
- Graphing Linear Inequalities and Systems of Inequalities quiz (10 questions)
- Graphing Linear Inequalities and Systems of Inequalities test (45 minutes)
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