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Hawkes Learning Systems:
College Algebra
3.5: Linear Inequalities in Two Variables
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Objectives
o Solving linear inequalities in two variables.
o Solving linear inequalities joined by “and” or “or”.
o Applications of the term regions of constraint.
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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.
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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.
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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).
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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).
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Solving Linear Inequalities in Two Variables
Step 1: Graph the line in ¡ 2 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.
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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 that half-plane solve
the inequality. Otherwise, the points in the
other half-plane solve the inequality.
Shade in the half-plane that solves the
inequality.
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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
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Example 1: 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 
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Example 2: Solving Linear Inequalities
Solve the following linear inequality by graphing its
solution set.
x y0
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Example 3: Solving Linear Inequalities
Solve the following linear inequality by graphing its
solution set.
x3
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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”.
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Example 4: 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
5 x  2 y  10
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Example 5: Linear Inequalities Joined by “And”
or “Or”
Graph the solution set that satisfies the following
x  y  4 or x  4
inequality.
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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.
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Example 6: Inequalities Involving Absolute
Values
Graph the solution set in ¡ 2 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
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Example 6: Inequalities Involving Absolute
Values (Cont.)
We now intersect the solution sets to obtain the final
answer:
x  3  1 and y  2  3
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Interlude: Regions of Constraint
o One noteworthy application of the ideas in this
section is linear programming, an important
mathematical tool in business and the social
sciences.
o Linear programming is a method used to maximize
or minimize a variable expression, subject to certain
constraints on the values of the variables.
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Interlude: Regions of Constraint
o The first step in linear programming is to determine
the region of constraint, a mathematical description
of all the possible values that can be taken on by the
variables (also called the feasible region).
o If the variable expression to be maximized or
minimized contains just two variables, the region of
constraint will be a portion of the Cartesian plane,
and it will usually be defined by the intersection of a
number of half-planes.
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Example 7: Regions of Constraint
A family orchard is in the business of selling peaches and
nectarines. The members of the family know that to prevent a
certain pest infestation, the number of nectarine trees in their
orchard cannot exceed the number of peach trees. Also, because
of the space requirements of each type of tree, the number of
nectarine trees plus twice the number of peach trees cannot
exceed 100 trees. Graph the region of constraint for this
situation.
Let p represent the number of peach trees and n the number of
nectarine trees. The following constraints are given:
n p
n0
n  2 p  100
p0
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Example 7: Regions of Constraint (Cont.)
Graph of the intersection of the four half-planes that
solve each individual inequality, with the horizontal axis
representing p and the vertical axis n :