Linear Inequalities: Estimations
One linear inequality with two unknowns
The linear inequality with unknowns #x# and #y# has the form \[ax+by+c\ge0\tiny.\]
In case you miss the sign, #\le#, remember that the equation is equivalent to (i.e., has the same solutions as) \[-ax-by-c\le0\tiny.\]
To determine which nodes in the plane meet these requirements, first draw the line #l# that is given by the equation #ax+by+c=0# .
All nodes with #ax+by+c\ge0# are on one side of the line #l# .
The line given by the equation #-43\cdot x+36\cdot y-427=0#, divides the plane into two areas, indicated as I and II in the figure below.
Which area is the solution of the inequality?
\[ -43\cdot x+36\cdot y-427 \gt 0 \]
Which area is the solution of the inequality?
\[ -43\cdot x+36\cdot y-427 \gt 0 \]
I
After all, the node #\rv{ -13 , {{32}\over{3}} }# lies in this area and the value of #-43\cdot x+36\cdot y-427# at this node has the sign #+#. Consequently area I satisfies the given inequality.
The solution can be found as follows: Node #\rv{ -13 , {{32}\over{3}} }# is in area I. The value of #-43\cdot x+36\cdot y-427# at this node has sign #+# . Consequently, #-43\cdot x+36\cdot y-427 \gt 0# is true in area I. If we move from this node to a node in area II, then the sign changes.
After all, the node #\rv{ -13 , {{32}\over{3}} }# lies in this area and the value of #-43\cdot x+36\cdot y-427# at this node has the sign #+#. Consequently area I satisfies the given inequality.
The solution can be found as follows: Node #\rv{ -13 , {{32}\over{3}} }# is in area I. The value of #-43\cdot x+36\cdot y-427# at this node has sign #+# . Consequently, #-43\cdot x+36\cdot y-427 \gt 0# is true in area I. If we move from this node to a node in area II, then the sign changes.
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