## Problem on sketching the region of a 2d inequality

Sketch the region in 2d satisfied by $( 1, 1 ) \cdot \bfx \geq 2.$
• ## Solution

To understand the region given by this inequality, we begin by drawing the region corresponding the equality $(1, 1 ) \cdot \bfx = 2.$ This will be the boundary of the region of interest.
Recall that
We observe that $(1, 1 ) \cdot \bfx = 2$ is a line with normal vector $( 1, 1 )$.
In order to sketch it, we need a point that it contains. That is, we need an $\bfx_0$ such that $(1, 1)\cdot \bfx_0 = 2$.
One such pair of values is $x_0 = 1, y_0=1$.
Hence, the boundary of the 2d region is given by the line with normal vector $\langle 1, 1\rangle$ that goes through the point $\bfx_0 = (1,1)$.
We could write the line in the form $(1, 1) \cdot \bfx = ( 1, 1) \cdot \bfx_0.$
The rest of the region is given by points such $$(1, 1) \cdot (\bfx - \bfx_0) \geq 0$$
Recall that
Hence the 2d region is given by the line and the half space in the direction of $(1, 1)$.