Graphing Linear Inequalities
You have probably already studied linear equations. We now turn
our attention to linear inequalities.
Definition
A linear inequality is a linear equation with the equal sign replaced by an inequality
symbol.
Linear Inequality
If A, B, and C are real numbers with A and B not both zero, then
Ax + By ≤ C is called a linear inequality. In place of
≤, we can also use
≥,
<, or >.
Graphing Linear Inequalities
Consider the inequality x + y > 1. If we solve the inequality for y, we get y > x + 1.
Which points in the xyplane satisfy this inequality? We want the points where the
ycoordinate is larger than the xcoordinate plus 1. If we locate a point on the
line y = x + 1, say (2, 3), then the ycoordinate is equal to the xcoordinate plus 1.
If we move upward from that point, to say (2, 4), the ycoordinate is larger than the
xcoordinate plus 1. Because this argument can be made at every point on the line, all
points above the line satisfy y > x + 1. Likewise, points below the line satisfy
y
< x + 1. The solution sets, or graphs, for the inequality y > x + 1 and the
inequality y
< x + 1 are the shaded regions shown in the figures below. In each case the line y
= x + 1 is dashed to indicate that points on the line do not
satisfy the inequality and so are not in the solution set. If the inequality symbol
is ≤ or ≥, then points on the boundary line also satisfy the inequality, and the line is
drawn solid.
Every nonvertical line divides the xyplane into two regions. One region is
above the line, and the other is below the line. A vertical line also divides the plane
into two regions, but one is on the left side of the line and the other is on the right
side of the line. An inequality involving only x has a vertical boundary line, and its
graph is one of those regions.
Graphing a Linear Inequality
1. Solve the inequality for y, then graph y = mx + b.
y > mx + b is satisfied above the line.
y = mx + b is satisfied on the line itself.
y < mx + b is satisfied below the line.
2. If the inequality involves x and not y, then graph the vertical line x = k.
x > k is satisfied to the right of the line.
x = k is satisfied on the line itself.
x
< k is satisfied to the left of the line.
Example 1
Graphing linear inequalities
Graph each inequality.
a)
b) y
≥  2x + 1
c) 3x  2y
< 6
Solution
a) The set of points satisfying this inequality is the region below the line
. To show this region, we first graph the boundary line
. The slope of
the line is
and the yintercept is (0,
1). Start at (0, 1) on the yaxis, then
rise 1 and run 2 to get a second point of the line. We draw the line dashed because
points on the line do not satisfy this inequality. The solution set to the inequality
is the shaded region shown in the figure below.
b) Because the inequality symbol is ≥, every point on or above the line satisfies
this inequality. To show that the line y =  2x + 1 is included, we make it a
solid line. See the figuer below.
c) First solve for y:
3x  2y 
< 6 
2y 
< 3x + 6 
y 

Divide by 2 and reverse the inequality. 
To graph this inequality, use a dashed line for the boundary
and shade
the region above the line. See the figure below for the graph.
Caution In Example 1(c) we solved the inequality for y before graphing
the line. We did that because <
corresponds to the region below the line and >
corresponds to the region above the line only when the inequality is solved for
y.
