Chapter ISG, Problem 6.6.13P

a.

To determine

The required inequalities for the given region.

Answer to Problem 6.6.13P

The required inequalities are-

  $x<0$ ; $y>0$

Explanation of Solution

Given: The shaded region consists of all points in the second quadrant exclusive of the axes.

Concept Used: 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: The second quadrant is a region bounded by the x and y axes.

Equation of the axes are-

  $y=0$ and $x=0$ respectively.

In the second quadrant every point has a negative x coordinate but a positive y coordinate. Hence the required set of inequalities that describes every point in the second quadrant excluding the axes is-

  $x<0$ ; $y>0$

b.

To determine

The required inequalities for the given region.

Answer to Problem 6.6.13P

The required inequalities are-

  $x<0$ ; $y<0$

Explanation of Solution

Given: The shaded region consists of all points in the third quadrant exclusive of the axes.

Concept Used: 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: The third quadrant is a region bounded by the x and y axes.

Equation of the axes are-

  $y=0$ and $x=0$ respectively.

In the third quadrant every point has negative x and y coordinates. Hence the required set of inequalities that describes every point in the third quadrant excluding the axes is-

  $x<0$ ; $y<0$

c.

To determine

The required inequalities for the given region.

Answer to Problem 6.6.13P

The required inequalities are-

  $x>0$ ; $y<0$

Explanation of Solution

Given: The shaded region consists of all points in the fourth quadrant exclusive of the axes.

Concept Used: 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: The fourth quadrant is a region bounded by the x and y axes.

Equation of the axes are-

  $y=0$ and $x=0$ respectively.

In the fourth quadrant every point has a positive x coordinate but a negative y coordinate. Hence the required set of inequalities that describes every point in the second quadrant excluding the axes is-

  $x>0$ ; $y<0$

d.

To determine

The required inequalities for the given region.

Answer to Problem 6.6.13P

The required inequalities are-

  $x>0$ ; $y\ge 0$

Explanation of Solution

Given: The shaded region consists of all points in the first quadrant, including the horizontal axis but excluding the vertical axis.

Concept Used: 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: The first quadrant is a region bounded by the x and y axes.

Equation of the axes are-

  $y=0$ and $x=0$ respectively.

In the first quadrant every point has positive x and y coordinates. Hence the required set of inequalities that describes every point in the first quadrant including the horizontal, i.e.- x axis but excluding the vertical, i.e.- y axis is-

  $x>0$ ; $y\ge 0$