There are many ways of finding the area of the required region. We will look at two of those methods and both those methods are pretty easy.
First Method:
First of all, to find the area of the triangle, we should know the base and height of that triangle. Height is clearly visible as it will be the y-intercept. So, the height of the given triangle is (5 units), and our objective should be to find out the base of the triangle. Base of the triangle is the X-axis line enclosed between the V-shaped modulus graphs. So, if we can find out the co-ordinates of the end-points of V-shaped modulus graph intersecting the X-axis, we can easily find out the distance between the two points which will serve our purpose.
Now, the value of x-coordinate is –1 and +9. Can we find out the distance between these two points which will give us the length of the triangle?
Yes, the distance is 9 – (–1) = 10 units and the height is 5 units.
So, area of the required area is = (1/2) × 10 × 5 = 25 square units.
Alternative method:
This is the far better efficient way of solving the question related with area problems. We just need to find out how much is the slope of the line, the perpendicular will bisect the base into two equal parts. Depending on what is the change in Y-coordinate and slope of the line, we can find out the two parts of base and add them to find the length of base of triangle.
We will have to count all integral points on X-axis, points on inverted V-shaped and all the points lying in the region enclosed between them since equality sign is included in question.
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