Graphing Inequalities in Two Variables
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Graphing Inequalities in Two Variables
Graphing Inequalities in Two Variables
Like a linear equation in two variables, the
solution set of an inequality in two variables is
graphed on a coordinate plane.
The solution set of an inequality in two variables is
the set of all ordered pairs that satisfy the
inequality.
Ordered Pairs that Satisfy an Inequality
From the set {(3, 3), (0, 2), (2, 4), (1, 0)}, which ordered
pairs are part of the solution set for
Use a table to substitute the x and y values of each
ordered pair into the inequality.
x
y
True or False
3
3
true
0
2
false
2
4
true
1
0
false
Ordered Pairs that Satisfy an Inequality
Answer: The ordered pairs {(3, 3), (2, 4)} are part of the
solution set of
. In the graph, notice
the location of the two ordered pairs that are
solutions for
in relation to the line.
Ordered Pairs that Satisfy an Inequality
From the set {(0, 2), (1, 3), (4, 17), (2, 1)}, which
ordered pairs are part of the solution set for
Answer: {(1, 3), (2, 1)}
Graphing Inequalities in Two Variables
The solution set for an inequality in two variables
contains many ordered pairs when the domain and
range are the set of real numbers.
The graphs of all these ordered pairs fill a region
on the coordinate plane called a half-plane.
An equation defines the boundary or edge for each
half-plane.
Graphing Inequalities in Two Variables
Graphing Inequalities in Two Variables
Consider the graph of y > 4.
First determine the boundary by
graphing y = 4, the equation
obtained by replacing the inequality
sign with an equals sign.
y
O
Since the inequality involves yvalues greater than 4, but not equal
to 4, the line should be dashed.
The boundary divides the coordinate plane into two
half-planes.
x
Graphing Inequalities in Two Variables
To determine which half-plane
contains the solution, choose a point
from each half-plane and test it in the
inequality.
Try (3, 0).
y > 4 (y = 0)
0 > 4 FALSE
Try (5, 6).
y > 4 (y = 6)
6 > 4 TRUE
The half-plane containing (5, 6), the
point that satisfies the inequality,
contains the solution. Shade that
half-plane.
y
O
graph of y > 4.
x
Graph an Inequality
Step 1 Solve for y in terms of x.
Original inequality
Add 4x to each side.
Simplify.
Divide each side by 2.
Simplify.
Graph an Inequality
Step 2 Graph
Since
does not include
values when
the
boundary is not included in the
solution set. The boundary
should be drawn as a dashed line.
Step 3 Select a point in one of
the half-planes and test it.
Let’s use (0, 0).
Original inequality
false
y = 2x + 3
Graph an Inequality
Answer: Since the statement is
false, the half-plane containing the
origin is not part of the solution.
Shade the other half-plane.
y = 2x + 3
Graph an Inequality
Answer: Since the statement is
false, the half-plane containing the
origin is not part of the solution.
Shade the other half-plane.
Check Test the point in the
other half-plane, for example,
(–3, 1).
Original inequality
Since the statement is true, the half-plane
containing (–3, 1) should be shaded. The
graph of the solution is correct.
y = 2x + 3
Graph an Inequality
Answer: