Question

Solve the following inequalities graphically. $2x+y\ge 4,x+y\le 3$ and $2x-3y\le 6$

Answer 1

We have , $2x+y\ge 4,...\left(i\right)$

$x+y\le 3...\left(ii\right)$

and $2x-3y\le 6...\left(iii\right)$

Converting these inequalities into equations, we have

$(2x+y=4,x+y=3$ and $2x-3y=6$

For inequality (i)

The line $2x+y=4$meets the coordinate axes at A(2,0) and B(0,4) respectively, Join the points (2,0) and (0,4) by a dark line, On putting (0,0)in the given inequality, which is not satisfying the inequality. $2x+y\ge 4$. so half plane of $2x+y\ge 4$ does not contain the origin.

For inequality (ii)

The line x+y=3 meets the coordinate axes at C(0,3) and D(3,0), Join the points (0,3) and (3,0) by dark line clarity , (0,0) satisfy the inequality$x+y\le 3.$ So, half plane of $x+y\le 3$ contains the origin.

For inequality(iii)

The line 2x-3y=6 meets the coordinates axes at D(3,0) and E(0,-2) Join the points (3,0) and (0,-2) by dark line clearly, (0,0) satisfy the inequality$2x-3y\le 6$ So half plane of $2x-3y\le 6$, contains the origin.

Now plot the inequalities on a graph paper which are shown below.

Hence, the shaded region gives required solution set.