Question

The graph of the inequality $2x + 3y > 6$ is
A. half plane that contains the origin.
B. half plane that neither contains the origin nor the points of the line $2x + 3y = 6$.
C. whole XOY – plane excluding the points on the line $2x + 3y = 6$.
D. entire XOY plane.

Answer 1

Hint: We will take the point of x in the equation then, we will get the points of y. Further, we will plot the points on a graph. Thereafter, we will find the answer from the graph


Complete step by step solution:
 The given inequality is
$2x + 3y > 6$
$3y > 6 - 2x$
$y > \dfrac{{6 - 2x}}{3}$
Let $y = \dfrac{{6 - 2x}}{3}\,.........\left( i \right)$
Putting $x = 0$ in equation (i), we have
$y = \dfrac{{6 - 2\left( 0 \right)}}{3}$
$y = \dfrac{{6 - 0}}{3}$
$y = \dfrac{6}{3}$
$y = 2$
$\left( {0,2} \right)$
Putting $x = 3$ in equation (i), we have
$y = \dfrac{{6 - 2\left( 3 \right)}}{3}$
$y = \dfrac{{6 - 6}}{3}$
$y = \dfrac{0}{3}$
$y = 0$
$\left( {3,0} \right)$
Putting $x = - 3$ in equation (i), we will get
$y = \dfrac{{6 - 2\left( { - 3} \right)}}{3}$
$y = \dfrac{{6 + 6}}{3}$
$y = \dfrac{{12}}{3}$
$y = 4$
$\left( { - 3,4} \right)$
Now, let us make a table to plot the point in graph

x	$0$	$3$	$ - 3$
y	$2$	$0$	$4$

Now, we putting the $\left( {0,2} \right),\left( {3,0} \right)$ and $\left( { - 3,4} \right)$ on the graph

Hence, the correct option (B).


Note: Students should carefully plot the points on the graph. If you are plotting wrong points on the graph then you will get the wrong answer.