Linear functions form the basis of linear inequalities. A linear
inequality in two variables relates two variables using an
inequality symbol, such as y > 2x – 4. Its graph is a region of
the coordinate plane bounded by a line. The line is a boundary
line, which divides the coordinate plane into two regions.
To graph y ≥ 2x – 4, make
the boundary line solid, and
shade the region above the
line. To graph
y > 2x – 4, make the
boundary line dashed
because y-values equal to 2x
– 4 are not included.
Helpful Hint
Think of the underlines in the symbols ≤ and ≥ as representing
solid lines on the graph.
Ex 1A: Graph the inequality
The boundary line is
a slope of
.
which has a y-intercept of 2 and
.
Draw the boundary line dashed because it is not part of the
solution.
Then shade the region above the boundary line to show
.
Check Choose a point in the solution region, such
as (3, 2) and test it in the inequality.
?
2 >? 1 
The test point satisfies the inequality, so the solution
region appears to be correct.
Ex 1B: Graph the inequality y ≤ –1.
Recall that y= –1 is a horizontal line.
Step 1 Draw a solid line for y=–1
because the boundary line is part of
the graph.
Step 2 Shade the region below the
boundary line to show where y < –1.
Check The point (0, –2) is a solution because
–2 ≤ –1. Note that any point on or below y = –1
is a solution, regardless of the value of x.
Ex 2: Graph 3x + 4y ≤ 12 using intercepts.
Substitute x = 0 and y = 0 into 3x + 4y = 12 to find the intercepts of
the boundary line.
x-intercept
3x + 4y = 12
3x + 4(0) = 12
3x = 12
x=4
(0, 3)
(4, 0)
y-intercept
(4, 0)
3x + 4y = 12
3(0) + 4y = 12
4y = 12
y=3
(0, 3)
Ex 3: A school carnival charges $4.50 for adults and $3.00 for
children. The school needs to make at least $135 to
cover expenses.
A. Using x as the adult ticket price and y as the child
ticket price, write and graph an inequality for the
amount the school makes on ticket sales.
B. If 25 child tickets are sold, how many adult tickets
must be sold to cover expenses?
An inequality that models the problem is 4.5x + 3y ≥ 135.
Find the intercepts of the boundary line.
4.5x + 3(0) = 135
x = 30
(30, 0)
4.5(0) + 3y = 135
y = 45
(0, 45)
If 25 child tickets are sold,
4.5x + 3(25) ≥ 135
4.5x + 75 ≥ 135
_
4.5x ≥ 60, so x ≥ 13.3
Substitute 25 for y in 4.5x + 3y ≥ 135.
Multiply 3 by 25.
A whole number of tickets must be sold.
At least 14 adult tickets must be sold.
You can graph a linear inequality that is
solved for y with a graphing calculator.
Press
and use the left arrow key
to move to the left side.
Each time you press
you will see
one of the graph styles shown here. You
are already familiar with the line style.
Ex 4: Solve
for y. Graph the solution.
8x – 2y > 8
–2y > –8x + 8
y < 4x – 4
Use the calculator option to shade below
the line y < 4x – 4.
Note that the graph is shown in the STANDARD
SQUARE window.
(
6:ZStandard followed by
5:ZSquare).