Solutions To Inequalities In Two Variables
In section ??? we discussed linear inequalities in one variable . A solution to
this type of
inequality was a value for the variable that made the inequality true. We now
wish to discuss
linear inequalities in two variables and solutions to these inequalities.
A solution to an inequality in two variables is an ordered pair that makes the
inequality true.
Example 1
Is 
a solution to
Solution
To de termine if is a solution to the given
inequality we will evaluate the inequality at this
point.
Which is a true statement, so
is a solution to
Example 2
Is 
a solution to
Solution
Again we will evaluate the inequality at the point
to see if it is a solution.
Which is false, so is not a solution to the
inequality 
Example 3
Is 
a solution to
Solution
Evaluating the inequality at the point we
have:
Which is a true statement, so
is a solution to
Note: In both examples 2 and 3 the ordered pair given is on the graph of the
line that we would
have by replacing the inequality with an “=”.
In other words the point
is on the graph of
the line 
and the point
is
on the graph of the line 
However, was not a
solution to its inequality and
was a solution. The
difference
came for the type of inequality that was used in each problem. In example 2 the
inequality use
was “<” and in example 3 the inequality was “≥”, it was the “or equal to” in the
inequality of
example 3 that allowed us to include the points on the line as part of the
solutions.
In either case it seems that the line is the dividing point to distinguish
solutions from
nonsolutions. This is exactly the situation. A line divides the coordinate
system into two halve ,
with an inequality one of those halves is the solution set.
Graphing Linear Inequalities in Two Variables
As mentions above a line divides the coordinate system into two halves, to
determine which half
is the solution set we shall select a point not on the line and see if it is a
solution to the
inequality. If is a solution we shall shade that side of the coordinate system
and if it is not a
solution we shall shade the side of the coordinate system not containing that
point. Remember
that the two sides we are referring to are determined by the line obtained by
replacing the
inequality with an “=”. Lets illustrate this further with an example.
Example 4
Graph the solution set of 
Solution
First we will graph the line
, this will divide the rectangular coordinate
system into
two halves, one of which will be our solution set. Since the inequality in this
problem is “<“, the
points on the line are not part of the solution set. To indicate this we shall
draw the line in
dashed instead of solid.
We now need to determine which side of the line we need to shade. To do this we
select a point
that is not on the line and determine whether or not it is a solution. We shall
select the point .
Since this is a true statement we shall shade the side of
the line that contains the point .
Putting this all together we have:
We now summarize these steps in the following .
| Steps To Graphing A Linear Inequality In Two
Variables |
1. Graph the line created by replacing the
inequality with an “=”.
The line should be dashed if the inequality is “<” or “>” and the line
should be
solid if the inequality is “≤” or “≥”
2. Select an ordered pair that is not on the line and determine whether
or not it is
a solution
3. If the point is a solution to the inequality, shade the half of the
coordinate
system containing that point.
If the point is not a solution then shade the half of the coordinate
system not
containing that point.
Remember: the line graphed in step 1 determines the halves of the
coordinate system. |
Example 5
Graph the solution set of 
Solution
First we graph the line 
If will be a solid
line since the inequality is “≤”. To make
the line easier to graph we shall solve the equation for y.
We will select
as our test point.
Checking to see if it is a solution to the inequality we
have:
which is a true statement, so
is a solution to the inequality, and we will shade the half of the
coordinate system that contains this point.
Example 6
Graph: 
Solution
We shall follow the same steps as in the previous examples . First we graph the
line 
. To
aide in the graphing we solve the equation for y .
The line will be dashed since the inequality is “>”. We
will select 
as our test point since the
line contains the origin. Testing this point yields:
Which is a false statement, so we will shade the side of the line that does not
contain the point
.
Problem Set Section 6.6
For questions 1 – 10 determine if the given point is a solution to the given
inequality.
For questions 11 – 30 graph the given inequality.
31. Below is the graph of the equation
, use it to graph the inequality
by
selecting a test point and using it to determine the appropriate part of the
coordinate system
to shade.
32. Below is the graph of the equation 
, use
it to graph the inequality 
by selecting a test point and using it to determine the appropriate part of the
coordinate system to shade.
