HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Hawkes Learning Systems:
College Algebra
3.5: Linear Inequalities in Two Variables
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Objectives
o Solving linear inequalities in two variables.
o Solving linear inequalities joined by “and” or “or”.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Linear Inequalities in Two Variables
If the equality symbol in a linear equation in two
variables is replaced with , , , or  , the result is a
linear inequality in two variables. A linear inequality
in the two variables x and y is an inequality that can
be written in the form
ax  by  c, ax  by  c, ax  by  c, ax  by  c,
Where a, b, and c are constants and a and b are not
both 0.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Linear Inequalities in Two Variables
o The solution set of a linear inequality in two variables
consists of all the ordered pairs in the Cartesian
plane that lie on one side of a line in the plane,
possibly including those points on the line.
o The first step in solving such an inequality is to
identify and graph this line.
o The line is simply the graph of the equation that
results from replacing the inequality symbol in the
original problem with the equality symbol.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Linear Inequalities in Two Variables
Any line in the Cartesian plane divides the plane into
two half-planes, and, in the context of linear
inequalities, all of the points in one of the two halfplanes will solve the inequality.
The green and
pink portions of
each graph are
the half-planes;
the blue lines
are the
boundary lines.
This graph has a closed half-plane (the This graph has an open half-plane (the
line is not included in the solution set).
line is included in the solution set).
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Linear Inequalities in Two Variables
The points on the line, called the boundary line in this
context, will also solve the inequality if the inequality
symbol is  or  , and this fact must be denoted
graphically by using a solid line.
The green and
pink portions of
each graph are
the half-planes;
the blue lines
are the
boundary lines.
This graph has a closed half-plane (the This graph has an open half-plane (the
line is not included in the solution set).
line is included in the solution set).
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Solving Linear Inequalities in Two Variables
Step 1: Graph the line in R2 that results from
replacing the inequality symbol with  .
Solid Line
Dashed Line
 or 
 or 
Non-strict.
Strict.
Points on the
line included in
the solution set.
Points on the line
excluded from the
solution set.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Solving Linear Inequalities in Two Variables
Step 2: Determine which of the half-planes
solves the inequality by substituting a test
point from one of the two half-planes into the
inequality. If the resulting statement is true, all
the points in the half-plane that contains the
test-point solve the inequality. If the resulting
statement is false, all the points in the other
half-plane that did not contain the test-point
solve the inequality. Shade in the half-plane
that solves the inequality.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Solving Linear Inequalities in Two Variables
Select a Test Point
Substitute into
the Inequality
true statement
Shade entire half-plane
that includes the test
point
false statement
Shade entire half-plane
that does not include
the test point
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example: Solving Linear Inequalities
Solve the following linear inequality by graphing its
solution set.
3 x  2 y  12
x-intercept:  4,0 
y-intercept:  0,6 
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example: Solving Linear Inequalities
Solve the following linear inequality by graphing its
solution set.
x y0
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example: Solving Linear Inequalities
Solve the following linear inequality by graphing its
solution set.
x3
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Solving Linear Inequalities Joined by “And” or
“Or”
o In Section 1.2, we defined the union of two sets A
and B , denoted A  B, as the set containing all
elements that are in set A or set B , and we defined
the intersection of two sets A and B , denoted A  B,
as the set containing all elements that are in both A
and B .
o For the solution sets of two inequalities A and B, A  B
represents the solution set of the two inequalities
joined by the word “or” and A  B represents the
solutions set of the two inequalities joined by the
word “and”.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example: Linear Inequalities Joined by “And” or
“Or”
To find the solution sets in the following problems, we will solve
each linear inequality individually and then form the union or
the intersection of the individual solutions, as appropriate.
Graph the solution set that satisfies the following inequalities.
5 x  2 y  10 and y  x
yx
5x  2 y  10
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example: Linear Inequalities Joined by “And” or
“Or”
Graph the solution set that satisfies the following
x  y  4 or x  4
inequality.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Inequalities Involving Absolute Values
o In Section 2.2, we saw that an inequality of the
form x  a can be rewritten as the compound
inequality a  x  a .
o This can be rewritten as the joint condition x  a
and x  a, so an absolute value inequality of this
form corresponds to the intersection of two sets.
o Similarly, an inequality of the form x  a can be
rewritten as x  a or x  a , so the solution of this
form of absolute value inequality is a union of two
sets.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example: Inequalities Involving Absolute
Values
Graph the solution set in R2 that satisfies the joint
conditions x  3  1 and y  2  3.
We need to identify all ordered pairs for which x  3  1
or x  3  1 while 3  y  2  3. That is, we need x  2
or x  4 while 1  y  5 . The solution sets of the two
conditions individually are:
x  2 or x  4
1  y  5
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example: Inequalities Involving Absolute
Values (Cont.)
We now intersect the solution sets to obtain the final
answer:
x  3  1 and y  2  3