how do i find the are of the region defined by |x|+|y| less than or equal to 5 and |x|+|y| greater than or equal to?
To find the area of the region defined by the inequalities |x| + |y| ≤ 5 and |x| + |y| ≥ 3, you can follow these steps:
Step 1: Understand the problem
The inequalities |x| + |y| ≤ 5 and |x| + |y| ≥ 3 define a region in the xy-plane. You want to find the area of this region.
Step 2: Sketch the region
To better visualize the region, it is helpful to sketch it on graph paper or using a graphing software. Start by drawing the coordinate axes (x-axis and y-axis), and then draw the lines |x| + |y| = 5 and |x| + |y| = 3.
Step 3: Determine the shape of the region
From the sketch, you will notice that the region defined by the inequalities is a square with rounded corners. It is known as a "diamond" or a "rhombus."
Step 4: Find the length of the sides
The rhombus has equal sides. To find the length of the sides, you can start by considering one of the vertices where the lines |x| + |y| = 5 and |x| + |y| = 3 intersect. These vertices lie on the coordinate axes, which means that either x or y will be zero. Plugging in x = 0 and solving for y in the equation |x| + |y| = 5 will give you one of the sides' length.
Step 5: Calculate the area
Since the rhombus has equal sides, you can then multiply the length of one side by the length of another side to find the area.
By following these steps, you should be able to find the area of the region defined by the given inequalities.