Start by finding a simple geometrical shape to enclose the required area.
Example System:
y≤ - 2 y≥ - 7 x ≥ 3 x ≤ 7
Practice makes perfect
We need our system of inequalities to lie on the fourth quadrant. Let's start by reviewing the quadrants. The coordinate plane is divided into four equal quarters called quadrants. The first quadrant is the one where x and y are positive, and the numbers increase counterclockwise.
Notice that the quadrant we are interested in is that where x takes positive values and y takes negative values.
A simple way to enclose the 20 units^2 is by using a quadrilateral. For example, if we choose a rectangle, the boundaries of the inequalities will all be horizontal and vertical lines which are of the form y=a and x=b respectively, with a and b being real numbers. The formula for the area is given below.
A= w * l
Since the area of a rectangle is obtained by multiplying its width w times its length l, we can just choose any two numbers with a product of 20 to set up its dimensions. For example, 4 and 5. Now we can position our rectangle anywhere in the fourth quadrant. Below we can find an example.
One of the vertical lines would have to pass through x=3 and another one through x=7. The horizontal lines will pass through y = -2 and y = - 7. Since we will be using vertical and horizontal lines , the equations for them will be of the same form. This is x =3, x =7, y=-2 and y=-7, respectively.
Finally, to contain the area of the rectangle we need the inequality related to y=- 2 to be shaded downwards. We can use y ≤ - 2. Similarly, we want to shade upwards from the line y= - 7, so we can use y ≥ - 7. Finally, we can use x ≥ 3 to shade to the right of the line x=3, and x ≤ 7 to shade to the left of the line x=7.
Our solution is the system of inequalities shown below.
y≤ - 2 y≥ - 7 x ≥ 3 x ≤ 7
Notice that this is only an example solution, as there are infinitely many ways to define a system of inequalities with the conditions required by this exercise. For example, we could have used another figure at first, or just different dimensions for the rectangle.
