Question

What is the shape of the graph of `|x|+|y|<1`?

Answer 1

The easiest thing is to just show you the graph

Explanation:

Answer 2

It is a diamond-shaped graph centred at the origin, with all 4 corners on the axes, 1 unit from the origin. The lines are dotted (to indicate "not equal to"), and the diamond is filled (to indicate "less than").

Explanation:

We're looking for all the points `(x, y)` where the sum of the horizontal and vertical displacements from the axes is less than 1.

Picture a taxi, picking someone up at the origin `(0,0)`. The passenger wants to go to `(0.6, 0.3)`, but the taxi can't go diagonally; it must move along the streets (gridlines). What is the shortest distance the taxi must travel?

It's easy; the taxi must go at least 0.6 units right, and 0.3 units up, for a total of 0.9 units of travel. It doesn't matter if the driver moves in a kind of "staircase" manner, they will still reach their destination after a total of 0.6 units right and 0.3 up. This means the point `(0.6,0.3)` has a taxicab distance of 0.9 from the origin. (Since 0.9 is less than 1, this point `(x,y)=(0.6, 0.3)` is an acceptable point on our graph.)

And it doesn't matter if the passenger wants to go to the left or down, either; what we're looking for is to keep the total distance from the origin less than 1, so all we need to do is find all points that are within a taxicab distance of 1 from the origin.

Here is a graph of `abs(x)+abs(y)<1`