Systems of Linear Inequalities
■ Sketch the graph of a linear inequality.
■ Sketch the graph of a system of linear inequalities.
LINEAR INEQUALITIES AND THEIR GRAPHS
The statements below are inequalities in two variables:
An ordered pair (a, b) is a solution of an inequality in x and y if the inequality is true when a and b are substituted for x and y, respectively. For example, (1, 1) is a solution of the inequality 3x-2y<6 because
The graph of an inequality is the collection of all solutions of the inequality. To sketch the graph of an inequality such as
begin by sketching the graph of the corresponding equation
The graph of the equation separates the plane into two regions. In each region, one of the following two statements listed below must be true.
1. All points in the region are solutions of the inequality.
2. No point in the region is a solution of the inequality.
So, you can determine whether the points in an entire region satisfy the inequality by simply testing one point in the region.
When possible, use test points that are convenient to substitute into the inequality, such as (0, 0).
Sketching the Graph of an Inequality in Two Variables
1. Replace the inequality sign with an equal sign, and sketch the graph of the resulting equation. (Use a dashed line for < or >and a solid line for ≤ or ≥.)
2. Test one point in each of the regions formed by the graph in Step 1. If the point satisfies the inequality, then shade the entire region to denote that every point in the region satisfies the inequality.
In this section, you will work with linear inequalities of the forms listed below.
The graph of each of these linear inequalities is a half-plane lying on one side of the line ax+by=c. When the line is dashed, the points on the line are not solutions of the inequality; when the line is solid, the points on the line are solutions of the inequality. The simplest linear inequalities are those corresponding to horizontal or Vertical lines, as shown in Example 1 on the next page.
EXAMPLE 1 Sketching the Graph of a Linear Inequality
Sketch the graph of each linear inequality.
a. x>-2 b. y≤3
a. The graph of the corresponding equation x=-2 is a vertical line. The points that satisfy the inequality x>-2 are those lying to the right of this line, as shown in Figure 1a.
b. The graph of the corresponding equation y=3 is a horizontal line. The points that satisfy the inequality y≤3 are those lying below (or on) this line, as shown in Figure 1b.
(a) Figure 1a,
b. Figure 1b,
EXAMPLE 2 Sketching the Graph of a Linear Inequality
Sketch the graph of x-y<2.
The graph of the corresponding equation x-y=2 is a line, as shown below. The origin (0, 0) satisfies the inequality, so the graph consists of the half-plane lying above the line. (Check a point below the line to see that it does not satisfy the inequality.)
For a linear inequality in two variables, you can sometimes simplify the graphing procedure by writing the inequality in slope-intercept form. For example, by writing x-y<2 in the form
you can see that the solution points lie above the line y=x-2.
SYSTEMS OF INEQUALITIES
Many practical problems in business, science, and engineering involve systems of linear inequalities. An example of such a system is shown below.
A solution of a system of inequalities in x and y is a point (x, y) that satisfies each inequality in the system. For example, (2, 4) is a solution of the above system because x=2 and y=4 satisfy each of the four inequalities in the system. The graph of a system of inequalities in two Variables is the collection of all points that are solutions of the system. For example, the graph of the above system is the region shown in Figure 2. Note that the point (2, 4) lies in the shaded region because it is a solution of the system of inequalities.
(2, 4) is a solution because it satisﬁes the system of inequalities.
To sketch the graph of a system of inequalities in two variables, first sketch the graph of each individual inequality (on the same coordinate system) and then find the region that is common to every graph in the system. This region represents the solution set of the system. For systems of linear inequalities, it is helpful to find the vertices of the solution region, as shown in Example 3.
EXAMPLE 3 Solving a System of Inequalities
Sketch the graph (and label the vertices) of the solution set of the system shown below.
You have already sketched the graph of each of these inequalities in Examples 1 and 2. The region common to all three graphs can be found by superimposing the graphs on the same coordinate plane, as shown below. To find the vertices of the region, find the points of intersection of the boundaries of the region.
Vertex A: (-2, -4)
Obtained by finding the point of intersection of
Vertex B: (5, 3)
Obtained by finding the point of intersection of
Vertex C: (-2, 3)
For the triangular region shown in Example 3, each point of intersection of a pair of boundary lines corresponds to a vertex. With more complicated regions, two border lines can sometimes intersect at a point that is not a vertex of the region, as shown at the right. To determine which points of intersection are actually vertices of the region, sketch the region and refer to your sketch as you find each point of intersection.
When solving a system of inequalities, be aware that the system might have no solution.
For example, the system
has no solution points because the quantity (x+y) cannot be both less than -1 and greater than 3, as shown below.
Another possibility is that the solution set of a system of inequalities can be unbounded. For example, consider the system below.
The graph of the inequality x+y<3 is the half-plane that lies below the line x+y=3. The graph of the inequality x+2y>3 is the half-plane that lies above the line x+2y=3. The intersection of these two half-planes is an inﬁnite wedge that has a vertex at (3, 0), as shown below. This unbounded region represents the solution set.
How Do You Formulate Inequalities from the Constraints?