2-5
Linear Inequalities in Two Variables
Objectives
Graph linear inequalities on the coordinate
plane.
Solve problems using linear inequalities.
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.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
For example, the line
y = 2x – 4, shown at
right, divides the
coordinate plane into
two parts: one where
y > 2x – 4 and one
where y < 2x – 4. In
the coordinate plane
higher points have
larger y values, so the
region where
y > 2x – 4 is above
the boundary line
where y = 2x – 4.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
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.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
Graph the inequality
The boundary line is
y-intercept of 2 and a slope of
Draw the boundary line
dashed because it is not
part of the solution.
Then shade the region above
the boundary line to show
.
Holt Algebra 2
.
which has a
.
2-5
Linear Inequalities in Two Variables
Example 1A Continued
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.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
Graph the inequality y ≤ –1.
Recall that y= –1 is a
horizontal line.
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.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
Graph the inequality y ≥ 3x –2.
The boundary line is y = 3x – 2 which has a
y–intercept of –2 and a slope of 3.
Draw a solid line because it
is part of the solution.
Then shade the region
above the boundary line to
show y > 3x – 2.
Check Choose a point in the solution region, such as
(0, 0) and test it in the inequality.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
If the equation of the boundary line is not in
slope-intercept form, you can choose a test point
that is not on the line to determine which region
to shade. If the point satisfies the inequality,
then shade the region containing that point.
Otherwise, shade the other region.
Helpful Hint
The point (0, 0) is the easiest point to test if it is
not on the boundary line.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
Graph 3x + 4y ≤ 12 using intercepts.
Step 1 Find the intercepts.
Substitute x = 0 and y = 0 into 3x + 4y = 12 to
find the intercepts of the boundary line.
y-intercept
3x + 4y
3(0) + 4y
4y
y
Holt Algebra 2
=
=
=
=
12
12
12
3
x-intercept
3x + 4y
3x + 4(0)
3x
x
=
=
=
=
12
12
12
4
2-5
Linear Inequalities in Two Variables
Step 2 Draw the boundary line.
The line goes through (0, 3) and
(4, 0). Draw a solid line for the
boundary line because it is part of
the graph.
(0, 3)
Step 3 Find the correct region
to shade.
Substitute (0, 0) into the
inequality. Because 0 + 0 ≤ 12
is true, shade the region that
contains (0, 0).
Holt Algebra 2
(4, 0)
2-5
Linear Inequalities in Two Variables
Graph 3x – 4y > 12 using intercepts.
(4, 0)
(0, –3)
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
Many applications of inequalities in two variables use only
nonnegative values for the variables. Graph only the part
of the plane that includes realistic solutions.
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?
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
Let x represent the number of adult tickets and y
represent the number of child tickets that must be sold.
Write an inequality to represent the situation.
Adult
price
4.50
times
number of
adult
tickets
•
x
plus
child
price
+
3.00
times
number
of child
tickets
is at
least
total.
•
y

135
An inequality that models the problem is 4.5x + 3y ≥ 135.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
Find the intercepts of the boundary line.
4.5(0) + 3y = 135
4.5x + 3(0) = 135
y = 45
Graph the boundary line
through (0, 45) and (30, 0)
as a solid line.
Shade the region above the
line that is in the first
quadrant, as ticket sales
cannot be negative.
Holt Algebra 2
x = 30
2-5
Linear Inequalities in Two Variables
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.
Holt Algebra 2
2-5
Linear Inequalities in Two Variables
ADVANCED LEVEL
Solve
for y. Graph the solution.
Solve 2(3x – 4y) > 24 for y. Graph the solution.
Holt Algebra 2