Chapter 6, Problem 25CCSR

To determine

Tofind:The system of inequalities for the given graph.

Answer to Problem 25CCSR

The system of inequalities for the given graph is

  $y\le 4x+3$

  $y>-3x-5$ .

Explanation of Solution

Given information:

The given graph is shown in figure here.

Methodused:

Find the equations of the inequalities using two point formula for line passing through two 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 through which the line/ boundary of graph of inequalities pass through are shown in Figure-2.

The equation of line AB that passes through $\text{A}\left(0,3\right)$ and $\text{B}\left(-2,-5\right)$ is

  $y-3=\frac{-5-3}{-2-0}\left(x-0\right)$

  $\Rightarrow y-3=4x$

  $\Rightarrow y=4x+3$

The equation of line CD that passes through $\text{C}\left(-3,4\right)$ and $\text{D}\left(-2,1\right)$ is

  $y-4=\frac{1-4}{-2-(-3)}\left(x-(-3)\right)$

  $\Rightarrow y-4=\frac{-3}{-2+3}\left(x+3\right)$

  $\Rightarrow y-4=-3x-9$

  $\Rightarrow y=-3x-5$

The inequality with boundary AB is either $y\le 4x+3$ or $y\ge 4x+3$ , we determine by a test point. See point $\left(1,3\right)$ is the solution of the inequality, so substituting this point in both inequalities, we have $3\le 4\times 1+3$ (true) and $3\ge 4\times 1+3$ (false).

Hence, the inequality with boundary AB is $y\le 4x+3$ .

The inequality with boundary CD is either $y<-3x-5$ or $y>-3x-5$ , we determine by a test point. See point $\left(-2,2\right)$ is the solution of the inequality, so substituting this point in both inequalities, we have $2<-3\left(-2\right)-5$ or $2<1$ (false) and $2>1$ (true).

Hence, the inequality with boundary CD is $y>-3x-5$

The graph and solution of obtained inequalities is shown in figure here.

Thus, the system of inequalities for the given graph is

  $y\le 4x+3$

  $y>-3x-5$

Conclusion:

The system of inequalities for the given graph is

  $y\le 4x+3$

  $y>-3x-5$ .