Chapter 6.6, Problem 25PPE

To determine

Towrite:The system of inequalities whose graph is given.

Answer to Problem 25PPE

The system of inequalities that represents given graph is $y\ge 2$ , $y>x+1$ .

Explanation of Solution

Given information:

The graph of system of inequalities is shown in Figure-1 here.

  

Methodused:

Find equations of lines (boundaries of shaded portions) and write inequalities corresponding to obtained lines.

The graphs of inequalities are shown in Figure-1.

A line parallel to x -axis is above origin at distance 2 units, so its equation is $y=2$ .

Since, inequality corresponding to this line shades the portion of coordinate plane $y\ge 2$ , hence the inequality with blue shaded region is

  $y\ge 2$..............(1)

Another line (dotted) makes intercepts of lengths -1 unit and 1 unit on x -axis and y -axis respectively so from formula $\frac{x}{a}+\frac{y}{b}=1$ is

  $\frac{x}{-1}+\frac{y}{1}=1$

  $\Rightarrow -x+y=1$

  $\Rightarrow y=x+1$.............. (2)

The point $\left(-1,2\right)$ lies in the shaded portion of inequality corresponding to equation (2), therefore, substituting $x=-1$ and $y=2$ in equation (2)

  $\Rightarrow 2=-1+1$

  $\Rightarrow 2>-1+1$

Therefore, corresponding inequality that represents yellow shaded region is

  $y>x+1$................. (3)

(Sign> is used as line is dotted, for solid line sign $\ge$ or $\le$ )

Verification:

The graph of obtained system of inequalities is

  

Hence, the system of inequalities that represents given graph is $y\ge 2$ , $y>x+1$ .

Conclusion:

The system of inequalities that represents given graph is $y\ge 2$ , $y>x+1$ .