In order to draw the graph of the inequality
(1), we take one point say (0, 0), in half plane I
and check whether values of
x
and
y
satisfy the
inequality or not.
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MATHEMATICS
We observe that
x
= 0,
y
= 0 satisfy the
inequality. Thus, we say that the half plane I is the
graph (Fig 6.7) of the inequality. Since the points on
the line also satisfy the inequality (1) above, the line
is also a part of the graph.
Thus, the graph of the given inequality is half
plane I including the line itself. Clearly half plane II
is not the part of the graph. Hence,
solutions
of
inequality (1) will consist of all the points of its graph
(half plane I including the line).
We shall now consider some examples to
explain the above procedure for solving a linear
inequality involving two variables.
Example 9
Solve 3
x
+ 2
y
> 6 graphically.
Solution
Graph of 3
x
+ 2
y
= 6 is given as dotted line in the Fig 6.8.
This line divides the
xy
-plane in two half
planes I and II. We select a point (not on the
line), say (0, 0), which lies in one of the half
planes (Fig 6.8) and determine if this point
satisfies the given inequality, we note that
3 (0) + 2 (0) > 6
or
0 > 6 ,
which is false.
Hence, half plane I is not the solution region of
the given inequality. Clearly, any point on the
line does not satisfy the given strict inequality.
In other words, the shaded half plane II
excluding the points on the line is the solution
region of the inequality.
Example 10
Solve 3
x
– 6
≥
0 graphically in
two dimensional plane.
Solution
Graph of 3
x
– 6 = 0 is given in the
Fig 6.9.
We select a point, say (0, 0) and substituting it in
given inequality, we see that:
3 (0) – 6
≥
0
or – 6
≥
0 which is false.
Thus, the solution region is the shaded region on
the right hand side of the line
x
= 2.
Fig 6.7
Fig 6.8
Fig 6.9
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LINEAR INEQUALITIES
127
Fig 6.10
Example 11
Solve
y
< 2 graphically.

Fig 6.11
EXERCISE
6.2
Solve the following inequalities graphically in two-dimensional plane:
1
.
x
+
y
< 5
2.
2
x
+
y
≥
6
3.
3
x
+ 4
y
≤
4
.
y
+ 8
≥
2
x
5.
x
–
y
≤
2
6.
2
x
– 3
y
> 6
7.
– 3
x
+ 2
y
≥
– 6
8.
3
y
– 5
x
< 30
9.
y
< – 2
10.
x
> – 3.
6.5
Solution of System of Linear Inequalities in Two Variables
In previous Section, you have learnt how to solve linear inequality in one or two variables
graphically. We will now illustrate the method for solving a system of linear inequalities
in two variables graphically through
some examples.
Example 12
Solve the following
system of linear inequalities graphically.
x
+
y
≥
5
... (1)
x
–
y
≤
3
... (2)
Solution
The graph of linear equation
x
+
y
= 5
is drawn in Fig 6.11.
We note that solution of inequality
(1) is represented by the shaded region
above the line
x
+
y
= 5, including the
points on the line.
On the same set of axes, we draw
the graph of the equation
x
–
y
= 3 as
shown in Fig 6.11. Then we note that inequality (2) represents the shaded region above
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12

128
MATHEMATICS
the line
x
–
y
= 3, including the points on the line.