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
ANSWERED
asked 2021-02-15
Determine the coordinate of vertices of the figure formed by a given system of inequalities.
\(\displaystyle{f{{\left({x}\right)}}}{\left\lbrace\begin{array}{c} {y}\ge-{4}\\{y}\le{2}{x}+{2}\\{2}{x}+{y}\le{6}\end{array}\right.}\)

Answers (1)

2021-02-16

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)
Therefore the graphical representation of given inequalities is shown below: From graph, it is clear that given system of inequalities from a triangle with vertices (1,4), (-3,-4) and (5,-4) R is region enclosed by guve system of inequalities.
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