Question

# Determine the coordinate of vertices of the figure formed by a given system of inequalities. f(x){(y>=-4),(y<=2x+2),(2x+y<=6):}

Inequalities systems and graphs
Determine the coordinate of vertices of the figure formed by a given system of inequalities.
$$\displaystyle{f{{\left({x}\right)}}}{\left\lbrace\begin{array}{c} {y}\ge-{4}\\{y}\le{2}{x}+{2}\\{2}{x}+{y}\le{6}\end{array}\right.}$$

Draw the line y = -4, now to obtain the region located by $$\displaystyle{y}\ge-{4}$$ take any point and substitute in given inequalities $$\displaystyle{y}\ge-{4}$$ if that point satisfies the inequality, then the region represented by that inequality must contain that point. Take (0,0) and $$\displaystyle{0}\ge-{4},\therefore \text{ the region represented by y }\ge-{4}$$ is region above the line y=-4
Draw the line y = 2x+2, now to obtain the region located by $$\displaystyle{y}\le{2}{x}+{2}$$ take any point and substitute in given inequalities $$\displaystyle{y}\le{2}{x}+{2}$$ if that point satisfies the inequality, then the region represented by that inequality must contain that point. Take (0,0) and $$\displaystyle{0}\le{2}\cdot{0}+{2}\Rightarrow{0}\le{2}$$, therefore the region represented by $$\displaystyle{y}\le{2}{x}+{2}$$ is region lies below line y=2x+2 andmust contain point (0,0)
Draw the line 2x+y = 6, now to obtain the region located by 2x+y$$\leq$$ 6 take any point and substitute in given inequalities $$2x+y \leq 6$$ if that point satisfies the inequality, then the region represented by that inequality must contain that point. Take (0,0) and $$\displaystyle{2}\cdot{0}+{0}\le{6}\Rightarrow{0}\le{6}$$, therefore the region represented by $$2x+y\leq 6$$ is region lies below line 2x+y=6 andmust contain point (0,0)