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Options would have helped.
Pick's theorem: A = I + P/2 - 1 where A is the area of the closed polygon, I is the number of internal integral points and P is the number of integral points on the boundary.
The figure forms a rhombus with diagonals 80 and 100. So the area is 1/2*80*100 = 4000.
Number of points on the line 4x + 5y = 200 which are not on the axes = 9
So, the number of points on the lines represented by 4|x|+5|y|=200 which are not on the axes are 9*4 = 36
Number of points on the axes are 4 => Total number of points on the line 4|x|+5|y|=200 are 36+4 = 40
So, according to the formula 4000 = I + (40/2) - 1 => I = 3981
There are 3981 points interior of the figure 4|x|+5|y| <= 200
But the points on the boundary are also to be considered because of the '=' sign
There are 40 boundary points
=> 3981 + 40 = 4021 integral solutions in total
