Write the inequality of the following graph

We can notice that : Line Passes through the Points (0 , 3) and (1 , 0)

Let us find the slope of the line in order to write the Equation of the Line.

Slope of a Line Passing through two points (x₁ , y₁) and (x₂ , y₂) is given by :

Points are (0 , 3) and (1 , 0)

here x₁ = 0 and x₂ = 1 and y₁ = 3 and y₂ = 0

As the line passes through (0 , 3) , The y - intercept is 3

We know that Equation of line is given by : y = mx + c

where m is the slope and c is the y-intercept

here m = -3 and c = 3

Equation of the Given line : y = -3x + 3

⇒ -3x - y = -3

⇒ 3x + y = 3

As the Given line is a Dotted line and the Shaded Region is Towards the Origin

The Inequality of the Graph is : 3x + y < 3

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!