Question

write a system of inequalities whose solution in the shaded region.....................

Answer 1

We have to identify the inequalities whose solution is shown as in the shaded region.

Identifying first inequality:

From the graph, (x1, y1)=(2, -2) and (x2, y2)=(-4, 0) is a point on the straight line.

The equation of the line can be derived using the formula for two point form as,

$\begin{gathered} y-y1=\frac{y2-y1}{x2-x1}(x-x1) \\ y-(-2)=\frac{0-(-2)}{-4-2}(x-2) \\ y+2=\frac{2}{-6}(x-2) \\ y+2=\frac{-1}{3}(x-2) \\ y=\frac{-1}{3}x+\frac{2}{3}-2 \\ y=\frac{-1}{3}x+\frac{2-2\cdot3}{3} \\ y=\frac{-1}{3}x+\frac{2-6}{3} \\ y=\frac{-1}{3}x-\frac{4}{3} \end{gathered}$

So, the equation of the line govering the inequality is y=-1/3x-4/3.

Since the line is a solid one, the symbol for the inequality is either ≤ or ≥.

Since the region below the line is shaded, the inequality should be of the form y≤.

So, one of the inequality is ,

$y\le-\frac{1}{3}x-\frac{4}{3}$

Identifying second inequality:

Now, the v shaped graph governing the inequality is of a modulus function shifted 4 units down.

Hence, the equation for this inequality graph can be written as,

$y=|x|-4$

Since the graph is a solid one, the symbol for the inequality is either ≤ or ≥.

Since the region above the graph is shaded, the inequality should be of the form y≥.

So, the inequality expression can be written as,

$y\ge|x|-4$

Therefore, the system of inequalities whose solution is in the shaded region is,

$\begin{gathered} y\le-\frac{1}{3}x-\frac{4}{3} \\ y\ge|x|-4 \end{gathered}$