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Home  >>  CBSE XI  >>  Math  >>  Linear Inequalities
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Solve the following system of equation graphically. $2x-y >1;x -2y <-1$

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Toolbox:
  • To represent the solution of linear inequality of one or two variable in a plane if the inequality involves $'\geq'$ or $' \leq$ we draw the graph of the line as a thick line to indicate the line is included in the solution set.
  • If the inequality involves $'>'$ as $'<'$ we draw the graph of the line using is not included in the solution set.
  • To solve an inequality $ax+by > c \qquad a \neq 0, b \neq 0 ( or \;> )$
  • We consider the corresponding equation $ax+by =c$ which represents a straight line This line divides the plane into two half planes I and II
  • We take any point in I half plane and check if it satisfies the given inequality will be one half plane (called solution region ) Containing the point satisfying the inequality
Step 1:
The first inequality is $2x-y >1$
Consider the equation $2x -y =1$
The points $(0,-1), (1,1)$ satisfy the equation.
The graph of the line using dotted lines is shown in the diagram.
The line divides the XY plane into two half planes.
Consider the point (0,0)
We see that $2(0) -0 >1$
$0 > 1$ is false.
Thus the region represented by the inequality $2x -y >1$ is the upper half plane which does not contain the point (0,0) [excluding the line 2x-y=1]
Step 2:
The second inequality is $x-2y <-1$
Consider the equation $x-2y=-1$
We see that the points $(-1,0) $ and $(1,1) satisfy the equation .
The graph of the line using dotted lines is shown in the diagram
Consider the point $(0,0)$
We see that $ 0-2(0) <-1$
$0 < -1$
is false
The region represented by the inequality $x-2y <-1$ is the region to the right of the line $x-2y=-1$ (excluding the line x-2y=-1)
Step 3 :
Hence the solution of the given system of linear inequalities is represented by the common shaded region, including the points on the lines.
answered Jul 31, 2014 by meena.p
 
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