When x=-5, |y|<4, so y<4 or -y<4, y>-4, making -4<y<4.
When x=-4 or 4, -5<y<5; x=-3 or 3, -6<y<6; x=-2 or 2, -7<y<7; x=-1 or 1, -8<y<8; x=0, -9<y<9.
The shape of the enclosed are is of two trapezoids joined by their longer base length 18, which is the distance between 9 and -9, when x=0. The base lies along the y axis. Because the graph is linear, we don't have to resort to calculus to work out the areas. Let's take the region to the right of the y axis. This is symmetrical about the x axis. The area of the trapezoid above the x axis to the right of the y axis is 4*5+4*4/2=28 sq units being the combined area of a rectangle and a triangle. The area below the x axis is the same, so on the positive side of x the area is 56 sq units.
For x between 0 and -5, we have the same area plus a trapezoidal strip of width one unit. The area of the strip is 9 sq units because it is made up of a rectangle of length 8 (between -4 and 4) and a square made up of two triangles at the ends of the trapezoid. The total area is 56+56+9=121 sq units.