Lesson 3.6
Core Focus on
Linear Equations
Graphing Linear Inequalities in
Two Variables
Warm-Up
Determine if each point is on the given line.
1. Is the point (2, 4) on the line y = −4x + 4?
No
2. Is the point (−1, 5) on the line 3x + 2y = 7?
Yes
3. Is the point (6, −2) on the line x = −2?
No
Lesson 3.6
Graphing Linear Inequalities
in Two Variables
Graph a linear inequality on a
coordinate plane.
Explore!
Step 1
In The Shade
Juniper is thinking of two numbers, x and y, whose
sum is at least 6.
a. Write an inequality that represents this statement.
b. Find 6 pairs of numbers Juniper may be thinking of.
List the sets of two numbers as ordered pairs (x, y).
Step 2 How many possible pairs of numbers fit Juniper’s
description? Explain your reasoning.
Step 3 On a coordinate plane, graph at least three ordered pairs that have a sum of
exactly six (remember that Juniper said the sum of her numbers is at least six).
Draw a solid line through these points. This is the boundary line for the
solution set. All values on this line have ordered pairs that have a sum of 6.
Step 4 Plot at least three of your ordered pairs from Step 1 on your graph. Are these
numbers “above” or “below” your boundary line? Is it possible for some of
your points to be on one side of the boundary line and other points on the
opposite side? Explain your reasoning.
Explore!
In The Shade
Step 5 Shade the coordinate plane on the side of the boundary line you placed your
ordered pairs on in Step 4. This shows all of the possible solutions to
Juniper’s description of her two numbers. Choose another point in your shaded
area as a test point. Do the x- and y-values add up to more than 6? If so, you
shaded the correct side of the boundary line.
Step 6 On a new coordinate plane, begin to graph the inequality y  x  1 by
2
1
graphing three points on the line y  x  1. DO NOT connect the points
2
with a solid line. Test one of the points on your line in your inequality. What
do you notice?
1
Step 7 Connect your points with a dashed line to show that these points are not
included in the solution. Choose a test point on either side of the boundary
line. Substitute the point into the original inequality. If the x- and y-values
make the statement true, shade that side of the boundary line. If they do not,
shade the side that does not contain your test point.
Step 8 Graph y ≤ −2x + 3 on a new coordinate plane.
Graphing Linear Inequalities
in Two Variables
1. Graph the boundary line by replacing the inequality sign with
an equal sign.
a. If the inequality sign is > or <, use a dashed boundary line.
b. If the inequality sign is ≥ or ≤, use a solid boundary line.
2. Choose a test point which is not on the boundary line.
Substitute the x- and y-values into the original inequality.
3. If the test point makes a true statement when substituted into
the inequality, shade the side of the boundary line that contains
that point. If the statement is false, shade the side of the
boundary line that does not contain the test point.
Example 1
Graph y > 3x − 4.
Graph the boundary line by graphing y = 3x – 4.
Use a dashed line to connect the points since the
inequality symbol is >.
Choose a test point that is not on the
boundary line.
(–2, 4)
The shaded region shows all
the possible ordered pairs that
are solutions to y > 3x – 4.
?
Substitute the coordinates into the original inequality
to see if the statement is true or false.
4  3(2)  4
Four is greater than negative 10 so the statement is true.
4  10
Shade the entire side of the boundary line that contains the test point (–2, 4).
Example 2
Graph x + 2y  4.
Rewrite the inequality as an equation. Convert
to slope-intercept form.
x + 2y = 4
–x
–x
2y = –x + 4
1
y = 2 x + 2
1
Graph the boundary line by graphing y =  2 x + 2.
Use a solid line to connect the points since the
inequality is .
Choose a test point that is not on the boundary line.
(4, 8)
Example 2 Continued…
Graph x + 2y  4.
?
Substitute the coordinates into the original inequality
to see if the statement is true or false.
4  2(8)  4
Twenty is NOT less than or equal to 4 so the
statement is FALSE.
20  4
Shade the entire side of the boundary line that
DOES NOT contain the test point (4, 8).
Communication Prompt
Boundary lines can be solid or dashed. What does each
type of line tell you about the inequality?
Exit Problems
Graph each linear inequality.
1. y > 2x − 1
5
3
2. y   x  4