Question

Check whether the half plane x + y + 4 > 0 contains the origin. Identify the plane and shade it.

Answer 1

Hint: To check whether the half plane x + y + 4 > 0 contains the origin or not, substitute the coordinates of origin (0, 0) in the equation of the plane and check if it satisfies the inequality or not. If it satisfies then the plane contains the given point otherwise not.

Complete step by step answer:
Here we have been provided with the plane x + y + 4 > 0 and we are asked to check whether the origin lies in this plane or not. Also we have to shade the plane in which the origin lies.
Now, the half plane x + y + 4 > 0 will contain the origin if the point of origin satisfies the given inequality of the plane. We know that the coordinates of the origin is (0, 0) so substituting the values of the abscissa and the ordinate in the given inequality we get,
\[\begin{align}
  & \Rightarrow 0+0+4>0 \\
 & \Rightarrow 4>0 \\
\end{align}\]
The above condition is true that means the half plane x + y + 4 > 0 contains the origin. Geometrically it can be shown as: -

Hence, the half plane x + y + 4 > 0 contains the origin, i.e. the point O (0, 0).

Note: Note that in case the origin would not have satisfied the given half plane then we would have directly considered that the other half plane which is x + y + 4 < 0 would have contained the origin without checking. The given plane is a 2 – D plane as it contains only two variables x and y axis.