Chapter 6.5, Problem 20P

Write an inequality that represents each graph.

To determine

To write: The inequality that represents given graph.

Explanation of Solution

Given information:

The given graph is shown in Figure-1.

Method/ Formula used:

Find the equation of line representing boundary of the graph.

Equation of lie passing through points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$ .

Calculations:

The points $\left(0,2\right)$ and $\left(-6,4\right)$ is $y-2=\frac{4-2}{-6-0}\left(x-0\right)$

  $\Rightarrow y-2=\frac{2}{-6}x$

  $\Rightarrow y=-\frac{1}{3}x+2$

  $\Rightarrow x+3y-6=0$ .

The shaded region lies on the right of the line $x+3y-6=0$ .

Hence, the inequalities that representing given graph is $x+3y-6\ge 0$ .

Test:

Suppose $\left(0,4\right)$ falls within the shaded region that must satisfy the inequality: $0+3\cdot 4-6\ge 0\Rightarrow 6>0$ .

The point $\left(4,0\right)$ does not fall within the shaded region that should not satisfy the inequality: $4+3\cdot 0-6\ge 0\Rightarrow -2>0$ not correct.

Conclusion:

The inequalities that representing given graph is $x+3y-6\ge 0$ .