Introduction
Solving a linear inequality in two variables is similar to
graphing a linear equation, with a few extra steps that
will be explained on the following slides. Remember that
inequalities have infinitely many solutions and all the
solutions need to be represented. This will be done
through the use of shading.
1
2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts
• A linear inequality in two variables has a half plane as
the set of solutions.
• A half plane is a region containing all points that has
one boundary, which is a straight line that continues in
both directions infinitely.
2
2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
• To determine the solution set, first graph the inequality
as a line.
• Sometimes the line or the boundary is part of the
solution; this means it’s inclusive. Inequalities that
have “greater than or equal to” (≥) or “less than or
equal to” (≤) symbols are inclusive.
• Use a solid line when graphing the solution to
inclusive inequalities.
3
2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
• Other times the line or boundary is NOT part of the
solution; in other words, it’s non-inclusive.
Inequalities that have “greater than” (>) or “less than”
(<) symbols are non-inclusive.
• Use a dashed line when graphing the solution to noninclusive inequalities.
• Either all the points above the line or all the points
below the line will be part of the solution. To find out
which side of the line contains the solutions, choose a
point that is clearly on one side of the line or the other.
4
2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
• Substitute the test point into the inequality.
• If the test point makes the inequality true, shade the
side of the line (half plane) that contains the test point.
If it does not make the inequality true, shade the
opposite side of the line. Shading indicates that all
points in that region are solutions.
5
2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
Graphing Equations Using a TI-83/84:
Step 1: Press [Y=] and arrow over to the left two times so that
the cursor is blinking on the “\”.
Step 2: Press [ENTER] two times for the greater than icon “
”
and three times for the less than icon “
”.
Step 3: Arrow over to the right two times so that the cursor is
blinking after the equal sign.
Step 4: Key in the equation using [X, T, Θ, n] for x.
Step 5: Press [WINDOW] to change the viewing window, if
necessary.
Step 6: Enter in appropriate values for Xmin, Xmax, Xscl, Ymin,
Ymax, and Yscl, using the arrow keys to navigate.
Step 7: Press [GRAPH].
6
2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
Graphing Equations Using a TI-Nspire:
Step 1: Press the home key.
Step 2: Arrow over to the graphing icon (the picture of the
parabola or the U-shaped curve) and press [enter].
Step 3: At the blinking cursor at the bottom of the screen, press
once the backspace key (a left facing arrow). A menu
pops up that gives choices for less than or equal to (≤),
less than (<), greater than (>), and greater than or equal
to (≥). Choose the appropriate symbol by using the
arrow keys to navigate to the desired symbol and press
the center button of the navigation pad. Alternatively,
enter the number that is associated with the symbol.
Step 4: Enter in the equation and press [enter].
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2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
Step 5: To change the viewing window: press [menu], arrow
down to number 4: Window/Zoom, and click the center
button of the navigation pad.
Step 6: Choose 1: Window settings by pressing the center
button.
Step 7: Enter in the appropriate XMin, XMax, YMin, and YMax
fields.
Step 8: Leave the XScale and YScale set to auto.
Step 9: Use [tab] to navigate among the fields.
Step 10: Press [tab] to “OK” when done and press [enter].
8
2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
Graphing a Linear Inequality in Two Variables
1. Determine the symbolic representation (write the inequality using
symbols) of the scenario if given a context.
2. Graph the inequality as a linear equation.
3. If the inequality is inclusive (≤ or ≥), use a solid line.
4. If the inequality is non-inclusive (< or >), use a dashed line.
5. Pick a test point above or below the line.
6. If the test point makes the inequality true, shade the half plane
that contains the test point.
7. If the test point makes the inequality false, shade the half plane
that does NOT contain the test point.
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2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
Standard Form of Linear Equations and Inequalities
• Linear equations can also be written as ax + by =
c, where a, b, and c are real numbers.
• Similarly, an inequality can be written in the same
form but with an inequality symbol (<, >, ≤, or ≥)
instead of an equal sign.
• To convert to slope-intercept form (y = mx + b),
solve the equation or inequality for y.
• Remember to switch the inequality symbol if you
multiply or divide by a negative.
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2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
Intercepts
• An intercept is the point at which the line
intersects (or intercepts) the x- or y-axis.
• You have dealt with the y-intercept, which is the
point at which the line intersects the y-axis. When
an equation is in slope-intercept form, y = mx + b,
b is the y-intercept.
• The general coordinates for the y-intercept are (0,
y). Notice the x-coordinate of the y-intercept is 0.
• To solve for the x-intercept in an equation, set x =
0 and solve for y.
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2.3.1: Solving Linear Inequalities in Two Variables
Key Concepts, continued
• The x-intercept is the point at which the line
intersects the x-axis.
• The general coordinates for the x-intercept are (x, 0).
• To solve for the x-intercept in an equation, set y =
0 and solve for x.
• You can plot a line using the intercepts. Find the xand y-intercepts and then connect the points.
• Plotting a line using the intercepts is helpful in
linear inequalities that are in context.
• Generally, linear inequalities in context have the
constraint that the variables can’t be negative. This
means the line will stop at the intercepts.
2.3.1: Solving Linear Inequalities in Two Variables
12
Common Errors/Misconceptions
• using a solid line for a non-inclusive inequality or a
dashed line for an inclusive inequality
• shading the region that makes the inequality false
• forgetting to switch the inequality symbol when
multiplying or dividing by a negative if converting to
slope-intercept form
• forgetting to shade the half plane
13
2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice
Example 1
Graph the solutions to the following inequality.
y>x+3
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2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 1, continued
1. Graph the inequality as a linear equation.
Since the inequality is non-inclusive, use a
dashed line.
y=x+3
To graph the line, plot the y-intercept first, (0, 3). Then
use the slope to find a second point. The slope is 1.
Count up one unit and to the right one unit and plot a
second point. Connect the two points and extend the
line to the edges of the coordinate plane.
15
2.3.1: Solving Linear Inequalities in Two Variables
16
2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 1, continued
2. Pick a test point above or below the line
and substitute the point into the
inequality.
Choose (0, 0) because this point is easy to substitute
into the inequality.
y>x+3
(0) > (0) + 3
0>3
This is false!
17
2.3.1: Solving Linear Inequalities in Two Variables
18
2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 1, continued
3. Shade the appropriate half plane.
Since the test point makes the inequality false, all
points on that side of the line make the inequality
false. Shade above the line instead; this is the half
plane that does NOT contain the point.
19
2.3.1: Solving Linear Inequalities in Two Variables
✔
2.3.1: Solving Linear Inequalities in Two Variables
20
Guided Practice: Example 1, continued
21
2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice
Example 3
A company that manufactures MP3 players needs to hire
more workers to keep up with an increase in orders.
Some workers will be assembling the players and others
will be packaging them. The company can hire no more
than 15 new employees. Write and graph an inequality
that represents the number of new workers who can be
hired.
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2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 3, continued
1. Create an inequality from the context.
There are two jobs to perform.
Let x = the number of workers who will assemble the
MP3 players.
Let y = the number of workers who will package the
MP3 players.
x + y ≤ 15
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2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 3, continued
2. Graph the inequality as a linear equation.
Since the inequality is inclusive, use a
solid line.
x + y = 15
To graph the line, convert the standard form of the
equation to slope-intercept form.
y = –x + 15
Plot the y-intercept first, (0, 15). Then use the slope
to find a second point. The slope is –1. Count down
one unit and to the right one unit and plot a second
point. Connect the two points and extend the line.
2.3.1: Solving Linear Inequalities in Two Variables
24
Guided Practice: Example 3, continued
3. Determine where to stop the line.
Stop the line at the intercepts because there cannot
be negative employees.
To find the y-intercept, look at the equation in slopeintercept form. The y-intercept is 15. The y-intercept
coordinates are (0, 15).
To find the x-intercept, use the standard form of the
equation and set y = 0.
x + y = 15
x + (0) =15
x = 15
2.3.1: Solving Linear Inequalities in Two Variables
25
Guided Practice: Example 3, continued
The coordinates of the x-intercept are (15, 0).
26
2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 3, continued
4. Pick a test point above or below the line
and substitute the point into the
inequality.
Choose (0, 0) because this point is easy to substitute
into the inequality.
x + y ≤ 15
(0) + (0) ≤ 15
0 ≤ 15
This is true!
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2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 3, continued
28
2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 3, continued
5. Shade the appropriate half plane.
Since the test point makes the inequality true, that
means all points on that side of the line make the
inequality true. Shade the half plane that contains the
test point.
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2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 3, continued
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14
12
10
8
6
4
2
-16
-14
-12
-10
-8
-6
-4
-2
0
-2
2
4
6
8
10
12
14
16
-4
-6
-8
-10
-12
-14
-16
2.3.1: Solving Linear Inequalities in Two Variables
30
Guided Practice: Example 3, continued
6. Reduce the shading to fit the context of
the problem and add labels.
Having negative employees doesn’t make sense.
Stop the shading at the x-axis, the y-axis, and the
boundary line so that the shading ends at
(0, 15) and (15, 0).
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2.3.1: Solving Linear Inequalities in Two Variables
Guided Practice: Example 3, continued
x + y ≤ 15
✔
2.3.1: Solving Linear Inequalities in Two Variables
32
Guided Practice: Example 3, continued
33
2.3.1: Solving Linear Inequalities in Two Variables