Chapter ISG, Problem 6.6.5P

To determine

The system of inequalities that correctly depicts the shaded region of the graph.

Answer to Problem 6.6.5P

The system of inequalities are-

  $y\ge 1$

  $2x+y\le 9$

  $y\le 3$

  $2x+y\ge -1$

Explanation of Solution

Given: The given shaded region is-

  

Concept Used: Equation of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by-

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

Consider a line $ax+by=c$ which passes through a point $(x_1,y_1)$ . The line divides the Cartesian plane into two parts. The two regions are symbolic of two different inequalities viz.

  $ax+by\ge c$ or $ax+by\le c$ , where the equality sign is taken if points on lying on the line are also considered. Any point in the shaded region of an inequality should satisfy the inequality otherwise the point doesn’t lie in the shaded region of the inequality or the inequality sign must be reversed to obtain the correct inequation for the region. This is used as a test to correctly determine the inequalities for a given shaded region.

Calculations: Naming the shaded region $ABCD$ and extending the line segments we get the following graph-

  

The equation of the line passing through points $A(1,1)$ and $B(4,1)$ is-

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

  $\Rightarrow y=1$

Since, (2,2) must lie in the shaded region the required inequality is-

  $y\ge 1$ (Since $2>1$ )

The equation of the line passing through points $B(4,1)$ and $D(3,3)$ is-

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

  $\Rightarrow y-1=-2x+8$

  $\Rightarrow 2x+y=9$

Since, (2,2) must lie in the shaded region the required inequality is-

  $2x+y\le 9$ (Since $2(2)+1<9$ )

The equation of the line passing through points $C(2,3)$ and $D(3,3)$ is-

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

  $\Rightarrow y=3$

Since, (2,2) must lie in the shaded region the required inequality is-

  $y\le 3$ (Since $2<3$ )

The equation of the line passing through points $A(1,1)$ and $C(2,3)$ is-

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

  $\Rightarrow y-1=2x-2$

  $\Rightarrow 2x+y=-1$

Since, (2,2) must lie in the shaded region the required inequality is-

  $2x+y\ge -1$ (Since $2(2)+2>-1$ )

Therefore the required set of inequalities that correctly depict the shaded region are-

  $y\ge 1$

  $2x+y\le 9$

  $y\le 3$

  $2x+y\ge -1$