3-3: Solving Systems of Linear Inequalities
Standard: 9.2.4.5
Goals/Objectives:
•Graph a system of linear inequalities.
APK
Graphing Linear Inequalities in Two Variables
A linear inequality relates two variables using an inequality symbol. The curve that separates the points of the
solution set from the points that are not in the solution set of a linear inequality is called a boundary line.
Graphing linear inequalities in two variables is similar to graphing inequalities with one variable.
First, determine the boundary line. This line can be graphed by using the slope and y-intercept or the intercept
method.
Next, determine if the boundary line is solid or dashed.
•If you see < or >, use a dashed line to graph the boundary line.
•If you see ≤ or ≥, use a solid line to graph the boundary line.
Finally, determine if the solution set is above or below the boundary line. Choose two points on the coordinate
plane, one above the boundary line and one below. Substitute the x- and y-values into the original inequality.
One point will give a true statement, and the other will give a false statement. Shade the side of the point that
gives the true statement.
Alternatively, you can shade the solution set of the inequality if it is solved for y (which would look like the
slope-intercept form).
•If you see y < or y ≤, shade below the line.
•If you see y > or y ≥, shade above the line.
Note, this only works if the inequality is solved for y.
Example 1
Graph 2x − y > 4 .
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The boundary line is 2x − y = 4 . This can be graphed using the intercept method. € line is dashed, meaning that the The boundary coordinates of the points on the line do not solve the system. The area below the boundary line is shaded. (4, –2) is a solution. (0, 0) is not a solution. NEW INFORMATION
A system of linear inequalities is a set of two or more linear inequalities with the same variables. The solution
of a system of linear inequalities is the overlapping region of the shaded solution region of each individual
linear inequality. Usually this represents and infinite set of points.
APPLICATIONS
Graph each system of inequalities.
Example 1
y ≥ 4x − 4
y ≥ 3x − 3
First, graph y ≥ 4 x − 4 . The boundary line is y = 4 x − 4 ,
with slope 4 and y-intercept –4. Graph the inequality
with a solid line, and shade above the line.
On€the same coordinate plane, graph€y ≥ 3x − 3. The
boundary line is y = 3x − 3, with slope 3 and y-intercept
–3. Graph the inequality with a solid line and shade
above the line.
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The overlapping region is the solution region.
Example 2
x+y >5
x − y < −3
First, graph x + y > 5 . The boundary line is y = −x + 5 ,
with slope –1 and y-intercept 5. Graph the inequality with a
dashed line, and shade above the line.
€ x − y < −3. The
On€the same coordinate plane, graph
boundary line is y = x + 3 , with slope 1 and y-intercept 3.
Graph the inequality with a dashed line and shade above the
line.
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The overlapping region is the solution region.
Example 3
7x < y −16
y ≤ −5x − 2
First, graph 7x < y −16 . The boundary line is
y = 7x +16 , with slope 7 and y-intercept 16. Graph
the inequality with a dashed line, and shade above the
line.
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On the same coordinate plane, graph y ≤ −5x − 2 .
The boundary line is y = −5x − 2 , with slope –5 and
y-intercept –2. Graph the inequality with a solid line
and shade below the line.
€ solution region.
The overlapping region is the
€
Example 4
2x + 2y ≤ 4
3x − y > 1
First, graph 2x + 2y ≤ 4 . The boundary line is
y = −x + 2 , with slope –1 and y-intercept 2. Graph
the inequality with a solid line, and shade below the
line.
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On the same coordinate plane, graph 3x − y > 1. The
boundary line is y = 3x −1, with slope 3 and yintercept –1. Graph the inequality with a dashed line
and shade below the line.
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APPLICATIONS
A system may involve more than two linear inequalities. The bounded region is a polygon.
Graph the system of linear inequalities and classify the figure created by the solution region.
Example 1
x ≥9
y ≥ −18
x ≤ 13
y ≤ −4
Graph the solid boundary lines x = 9 and x = 13 and
shade the area between them. Graph the solid boundary
lines y = −18 and y = −4 and shade the area between
them. Since the horizontal lines are perpendicular to the
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vertical lines, the resulting solution region is a rectangle.
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Example 2
y≤7
2x − y ≤ 3
x + 2y ≥ −6
Graph the solid boundary lines y = 7 and shade below it.
Graph the solid boundary line 2x − y = 3 and shade
above it. Then graph the solid boundary line x + 2y = −6
and shade above it. The solution region is a triangle.
€
1
Since the slope of 2x€− y = 3 is and the slope of
2
€
x + 2y = −6 is –2, the lines are perpendicular
and the
triangle is a right triangle.
€
€
Example 3
x ≤ −1
y ≤ 3x + 2
y ≥ −3x −10
Graph the solid boundary lines x = −1 and shade to the left
of it. Graph the solid boundary line y = 3x + 2 and shade
below it. Then graph the solid boundary line y = −3x −10
and shade above it. The
€ solution region is an isosceles
triangle.
€
€
Example 4
y≥x
y ≤ x +6
x ≤6
x ≥ −2
Graph the solid boundary line y = x and shade above it.
Graph the solid boundary line y = x + 6 and shade below it.
Graph the solid boundary line x = 6 and shade to the left of
it. Graph the solid boundary line x = −2 and shade to the
€ region is a quadrilateral. Since the
right of it. The resulting
€ the opposite sides of the
boundary lines that form
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quadrilateral have the same slope, the figure is a
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parallelogram.
Application Problem
Fund-raising A charity is selling T-shirts in order to raise money. The cost of a T-shirt is $15 for adults and
$10 for students. The charity needs to raise at least $3000 and has only 250 T-shirts. Write and graph a system
of inequalities that can be used to determine the number of adult and student T-shirts the charity must sell.
Let A = the number of adult T-shirts sold, and let S = the number of student T-shirts sold.
First, write the inequality that describes how many T-shirts the charity can sell. The charity only has 250 Tshirts. The number of adult t-shirts plus the number of student t-shirts is less than or equal to 250.
A + S ≤ 250
Next, write the inequality that describes how
much money the charity would like to make. The
cost of each adult T-shirt is $15 and the cost of
each student T-shirt is $10. The charity would
like to make at least $3000.
The money made from the sales of adult t-shirts is
$15 times the number sold.
15A
The money made from the sales of student t-shirts
is $10 times the number sold.
10S
The money made from both types of t-shirts needs to be equal to, or more than, $3000.
15A +10S ≥ 3000
Both inequalities are shaded on the graph (above). The overlapping purple region represents the ordered pairs
of (adult t-shirts, student t-shirts) that satisfy both constraints. Note: Since both A and S are discrete variables
(meaning that it is not possible to buy or sell parts of a t-shirt), the solution set is finite, not infinite as in the
other examples.
SUMMARY
•Know how to solve a system of linear inequalities by graphing.
•Know how to determine the shape of a bounded region.
Other skills needed: graphing linear functions