In this lesson we graph the solutions to linear inequalities in two variables, and to systems of linear inequalities. We also find the vertices of the solution set. These skills are used when solving two-dimensional optimization problems subject to a set of constraints.
Begin by asking the class: How is an inequality in two variables different from an equation? How are the solutions different? How are their graphs different? (Remember that a graph is just a picture of all the solutions.)
Activity2.5.1.Graphing a Linear Inequality.
Graph the solutions to the linear inequality
\begin{equation*}
3x - y \le -2
\end{equation*}
The first step is to graph the equation \(3x - y = -2\text{.}\) This line will be the boundary line for the solution set.
Write the equation \(3x - y = -2\) in slope-intercept form.
Use the slope-intercept method to graph the line.
\(b = \)
\(m = \dfrac{\Delta y}{\Delta x} = \)
Next, use a test point to locate the solutions of the inequality. (We can use any point for a test point, as long as it does not lie on the line!) We’ll use \((1,4)\text{.}\) Label this point on your graph. Use algebra to decide whether \((1,4)\) is a solution of the inequality \(3x - y \le -2\text{.}\)
Which side of the line includes all the solutions of the inequality? Shade that side of the line. Is the line itself included in the solution set?
Can you suggest an easier test point instead of \((1,4)\text{?}\)
Graph the solutions to each inequality. (How do you know whether to make your boundary line solid or dashed?)
1. \(~~y - 3x \lt 6\)
2. \(~~y \gt \dfrac{-3}{2} x\)
Activity2.5.2.Systems of Inequalities.
To graph the solution set of a system, graph each inequality separately, one at a time. Shade each solution region lightly (or use different colors) so that you can see their intersection.
Graph each inequality in the system:
\begin{gather*}
5x + 4y \lt 40\\
-3x + 4y \lt 12\\
x \lt 6,~ y \gt 2
\end{gather*}
Inequality 1: \(~~5x + 4y \lt 40\)
\(x\)
\(y\)
\(0\)
\(\hphantom{00000}\)
\(\hphantom{00000}\)
\(0\)
Test Point:
Inequality 2: \(~~-3x + 4y \lt 12\)
\(x\)
\(y\)
\(0\)
\(\hphantom{00000}\)
\(\hphantom{00000}\)
\(0\)
Test Point:
Inequality 3: \(~~x \lt 6\)
Remember that \(x=6\) is a vertical line. Which side should you shade?
Inequality 4: \(~~y \gt 2\)
Remember that \(y=2\) is a horizontal line. Which side should you shade?
Outline the intersection of the four solution sets. Draw dots at the vertices, or corners, of the intersection region. (There are four vertices.)
Activity2.5.3.Finding the Vertices.
Use a system of equations to find the coordinates of each of the vertices of the solution set in Activity 2.
Intersection of Line 1 and Line 2:
Intersection of Line 1 and Line 3:
Intersection of Line 2 and Line 4:
Intersection of Line 3 and Line 4:
Graph the solutions to the system of inequalities, and find the vertices of the solution set.
