Graphical Solution of Linear Inequalities in Two Variables

Introduction to Graphical Solutions

A linear inequality in two variables, such as $ax+by < c$ or $ax+by \ge c$, represents a region in the Cartesian plane. The line $ax+by=c$ is the boundary line that divides the plane into two half-planes. The solution to the inequality is one of these half-planes.

Method for Graphing an Inequality

1. Draw the Boundary Line: First, replace the inequality symbol with an equals sign to get the equation of the boundary line, $ax+by=c$. Draw this line on the Cartesian plane.

   • If the inequality is slack ($\le$ or $\ge$), draw a solid line to indicate that the points on the line are included in the solution.
   • If the inequality is strict ($<$ or $>$), draw a dashed line to indicate that the points on the line are not included.

2. Choose a Test Point: Select any point that is not on the boundary line. The origin, (0,0), is the easiest choice if the line does not pass through it.

3. Shade the Solution Region: Substitute the coordinates of the test point into the original inequality.

   • If the inequality is true for the test point, shade the entire half-plane that contains the test point.
   • If the inequality is false, shade the other half-plane (the one that does not contain the test point).

The shaded region is the graphical representation of the solution set for the inequality.

Example 1: Basic Inequality (Solid Line)

Question: Solve the inequality $3x+4y \le 12$ graphically.

Solution:

1. Draw the Boundary Line: The equation is $3x+4y=12$. We draw a solid line through the points (0,3) and (4,0) because the inequality is $\le$.
2. Choose a Test Point: We use the origin (0,0).
3. Test and Shade: Substitute (0,0) into the inequality: $3(0)+4(0) \le 12 \implies 0 \le 12$. This is true.
4. Conclusion: We shade the half-plane that contains the origin. This region, including the line itself, is the solution.

Example 2: Strict Inequality (Dashed Line)

Question: Solve $x-y > 1$ graphically.

Solution:

1. Draw the Boundary Line: The equation is $x-y=1$. We draw a dashed line through (0,-1) and (1,0) because the inequality is strict ($>$).
2. Choose a Test Point: We use the origin (0,0).
3. Test and Shade: Substitute (0,0) into the inequality: $0-0 > 1 \implies 0 > 1$. This is false.
4. Conclusion: We shade the half-plane that does not contain the origin.

Example 3: Line Through the Origin

Question: Solve $y > -2x$ graphically.

Solution:

1. Draw the Boundary Line: The equation is $y=-2x$. This line passes through the origin (0,0). Draw a dashed line.
2. Choose a Test Point: Since the line passes through (0,0), we must choose another point. Let's pick (1,1).
3. Test and Shade: Substitute (1,1) into the inequality: $1 > -2(1) \implies 1 > -2$. This is true.
4. Conclusion: We shade the half-plane that contains the point (1,1).