↧↧↧ Detailed Video Solution to the Problem ↧↧↧Given that S be the set of all points (x, y) in the xy plane such that |x| + |y| ≤ 2 and |x| ≥ 1 and we need to find what is the area of the region represented by SPlotting |x| + |y| ≤ 2We will have four case for values of x and y each being ≥ 0 and ≤ 0 respectively (in four quadrants)
1. x≥0, y≥0 => |x| = x and |y| = y => x + y ≤ 2 (Quadrant 1). We will plot the line x + y = 2 and will consider the area below the line as we have ≤ sign
[Note: we need to keep the sign of y positive while considering the sign of the inequality]
2. x≤0, y≥0 => |x| = -x and |y| = y => -x + y ≤ 2 (Quadrant 2). We will plot the line -x + y = 2 and will consider the area below the line as we have ≤ sign
3. x≤0, y≤0 => |x| = -x and |y| = -y => -x - y ≤ 2 (Quadrant 3) => x + y ≥ -2. We will plot the line x + y = -2 and will consider the area above the line as we have ≥ sign
4. x≥0, y≤0 => |x| = x and |y| = -y => x - y ≤ 2 (Quadrant 4) => y ≥ x - 2. We will plot the line x - y = 2 and will consider the area above the line as we have ≥ sign
=> We get the square as the enclosed figure shown in the image below
Attachment:
Mod x + Mod y LTEQ 2.jpg [ 32.92 KiB | Viewed 2972 times ]
Plotting |x| ≥ 1We will have two cases x≥0 and x<0
1. x≥0 => |x| = x => x ≥ 1 and Intersection of x≥0 and x ≥ 1 equals x ≥ 1 => Area will be to the right of x = 1
2. x<0 => |x| = -x => -x ≥ 1 => x ≤ - 1 and Intersection of x<0 and x ≤ - 1 equals x ≤ - 1 => Area will be to the left of x = -
1
Attachment:
Mod x + Mod y LTEQ 2 intersection with mod x GTEQ 1.jpg [ 21.96 KiB | Viewed 2897 times ]
Point of intersection of both the graphsThere will be four points of intersections1. x=1 and x+y=2 will intersect at (1,1)
2. x=-1 and -x+y=2 will intersect at (-1,1)
3. x=-1 and -x-y=2 will intersect at (-1,-1)
4. x=1 and x-y=2 will intersect at (1,-1)
So, common area between the two graphs will be given by the two shaded triangles shown below:
Attachment:
Common area.jpg [ 23.42 KiB | Viewed 2835 times ]
Area of each triangle = \(\frac{1}{2}\) * Base * Height = \(\frac{1}{2}\) * 2 * 1 = 1
=> Total area of two triangles= 2*1 = 2 units
So,
Answer will be CHope it helps!
Watch the following video to learn Basics of Absolute Values