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# Solve the given inequality in two-dimensional plane . $x+y < 5$

Toolbox:
• Same Quantity can be added (a subtracted ) to (from ) both sides of the inequality with out changing the sign of the in equality.
• Same positive quantities can be multiplied or divided to both side of the in equality with out changing the sign of the inequality.
• If same negative quantity is multiplied or divided to both sides of the inequality is reversed i.e $'>'$ sign changes to $'<'$ and $'<'$ changes $'>'$ .
• 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
The given inequality is $x+y < 5$ consider the equation .
$x+y <5$
$(5,0)$ and $(0,5)$ satisfy the equation
The line $x+y =5$ is represented as a dotted line in the graph.
The line divides the plane into two half planes I and II.
Select a point not an line say (0,0)
Step 2:
We observe that , $0+0 <5$
$0 < 5$
Which is true (0,0) satisfies the given inequality
$\therefore$ half plane II is not the solution region.
Step 3:
The solution region of the given inequality is the shaded half plane I. not including the line.