Linear Inequality in Two Variables
An inequality of the form,
(i) $ax+by<c$
(ii) $ax+by\leq c$
(iii) $ax+by>c$
(iv) $ax+by\geq c$
where a,b,c are real numbers and $a,b\neq 0$ is called a linear inequality in two variables x and y.
Half Planes
A line divides the Cartesian plane into two parts. Each part is known as half plane.
(i) Left and Right half planes
A vertical line will divide the plane in left and right half planes.
(ii) Lower and Upper half planes
A non-vertical line will divide the plane into lower and upper half planes.
To find the graphical solution of an inequality in two variables
step 1: Consider the equation ax+by=c, which represents a straight line in xy-plane.
step 2: Put x=0 in the equation of step 1 to get value of y. Similarly, put y=0, to obtain the value of x.
step 3: Draw a line joining the points obtained in step 2 (x,0) (0,y). If the inequality is of the form $<$ or $>$, draw the dotted line. If the inequality is of the form $\leq$ or $\geq$, draw a thick line
step 4: Take any point not lying on the line (preferable origin (0,0)) and check whether this satisfies the given linear inequality or not.
step 5: If the inequality is satisfied, then shade the portion of the plane which contains the chosen point, otherwise shade the portion which does not contain the chosen point.
Example 1: $3x+4y\geq 12$
step 1: $3x+4y=12$
step 2: (i) put x=0,
$3(0)+4y=12$
$y=3$
(ii) put y=0,
$3x+4(0)=12$
$x=4$
step 3: Draw the line with values (0,3) and (4,0). Since inequality is $\geq$, draw thick line.
Consider origin (0,0)
$3(0)+4(0)=0\geq 12$, which is not true
step 4: Since (0,0) does not satisfy the given inequality, shade the portion which does not contain (0,0)
Example 2: $3x-4y\leq 12$
step 1: $3x-4y=12$
step 2: (i) put x=0,
$3(0)-4y=12$
$y=-3$
(ii) put y=0,
$3x-4(0)=12$
$x=4$
step 3: Draw the line with values (0,-3) and (4,0). Since inequality is $\leq$, draw thick line.
Consider origin (0,0)
$3(0)-4(0)=0\leq 12$, which true
step 4: Since (0,0) satisfies the given inequality, shade the portion which contain (0,0)
.
An inequality of the form,
(i) $ax+by<c$
(ii) $ax+by\leq c$
(iii) $ax+by>c$
(iv) $ax+by\geq c$
where a,b,c are real numbers and $a,b\neq 0$ is called a linear inequality in two variables x and y.
Half Planes
A line divides the Cartesian plane into two parts. Each part is known as half plane.
(i) Left and Right half planes
A vertical line will divide the plane in left and right half planes.
(ii) Lower and Upper half planes
A non-vertical line will divide the plane into lower and upper half planes.
step 1: Consider the equation ax+by=c, which represents a straight line in xy-plane.
step 2: Put x=0 in the equation of step 1 to get value of y. Similarly, put y=0, to obtain the value of x.
step 3: Draw a line joining the points obtained in step 2 (x,0) (0,y). If the inequality is of the form $<$ or $>$, draw the dotted line. If the inequality is of the form $\leq$ or $\geq$, draw a thick line
step 4: Take any point not lying on the line (preferable origin (0,0)) and check whether this satisfies the given linear inequality or not.
step 5: If the inequality is satisfied, then shade the portion of the plane which contains the chosen point, otherwise shade the portion which does not contain the chosen point.
Example 1: $3x+4y\geq 12$
step 1: $3x+4y=12$
step 2: (i) put x=0,
$3(0)+4y=12$
$y=3$
(ii) put y=0,
$3x+4(0)=12$
$x=4$
step 3: Draw the line with values (0,3) and (4,0). Since inequality is $\geq$, draw thick line.
Consider origin (0,0)
$3(0)+4(0)=0\geq 12$, which is not true
step 4: Since (0,0) does not satisfy the given inequality, shade the portion which does not contain (0,0)
Example 2: $3x-4y\leq 12$
step 1: $3x-4y=12$
step 2: (i) put x=0,
$3(0)-4y=12$
$y=-3$
(ii) put y=0,
$3x-4(0)=12$
$x=4$
step 3: Draw the line with values (0,-3) and (4,0). Since inequality is $\leq$, draw thick line.
Consider origin (0,0)
$3(0)-4(0)=0\leq 12$, which true
step 4: Since (0,0) satisfies the given inequality, shade the portion which contain (0,0)
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