Find the number of lattice points in the region defined by |x| + |y| < 5.
See the graph here : https://www.desmos.com/calculator/9riuxf41mb
We have 16 lattice points above and below the x axis and 9 points on the x axis....so....the total number of lattice points =
16 + 16 + 9 =
The strategy here is to find the boundary of the graph and then figure out what the actual graph is. First, plot some points for |x| + |y| = 5 on the Cartersian plane; (5,0), (4,1), (3,2), and combinations of their opposites work as well (ex. -4,1).
The graph of |x| + |y| = 5 becomes a diamond centered at the origin with perpendicular diagonals of length 10 on the x and y-axis.
If you think about it, |x| + |y| < 5 must be all of the area within the diamond.
At this point, you can count the number of lattice points by drawing a diagram. Be careful, as the inequality sign is less than, not less than or equal to, so do not count the lattice points on the diamond itself.
By counting with symmetry, you get
6*4 + 4*4 + 1 = 41.
The 6*4 is for the points between the axes, the 4*4 is the points on the axes (excluding the origin), and the 1 is for the origin.