How do you graph x + y < - 4 ?

Step 1
Although not the case, just for now consider it as ${\displaystyle x+y=-4}$ to determine the line to be graphed.
Subtract x from both sides
${\displaystyle y=-x-4}$
This will plot the line. However what we really have is:
${\displaystyle x+y<-44}$
Subtract x from both sides
${\displaystyle y<-x-4}$
As y is 'less than' something done to x, the value of y can never actually be the value on the line. Consequently we use a dotted line to signify this condition. You would also use a dotted line for greater than.
To indicate that y can take on any value less then the line the area under the line (less than) is shaded.