Chapter 3.3, Problem 46HP

a.

To determine

Find a system of inequalities.

Answer to Problem 46HP

  $x<0;y<0$

Explanation of Solution

Given information:

The solution lies in the third quadrant.

Calculation:

Confining the first inequality to just quadrant $2and3$ (to the left of $y-axis$ ), we get

  $x<0$

Again, confining the second inequality to just quadrant $3and4$ (below $x-axis$ ), we get

  $y<0$

Together, these inequalities define a space strictly contained in quadrant $3$ . Any system that confines a region strictly below $x-axis$ the and to the left $y-axis$ of will define a region in quadrant $3$ . So we get,

  $x<0;y<0$

Hence, the inequality of a solution that lies in the third quadrant is $x<0;y<0$ .

b.

To determine

Find a system of inequalities.

Answer to Problem 46HP

  $y\ge 2x+5;y\le 2x+1$ .

Explanation of Solution

Given information:

The solution that does not exist.

Calculation:

The inequality that defines a region lying above and along the line $y=2x+5$ is,

  $y\ge 2x+5$

The inequality that defines a region lying below and along the line $y=2x+1$ is,

  $y\le 2x+1$

These lines are parallel with slope. One region lies above the upper line, while the second region lies below the lower line. So, these regions have no common points and also, no solution. So we get this region as,

  $y\ge 2x+5;y\le 2x+1$

Hence, the inequality of a solution that does not exist is $y\ge 2x+5;y\le 2x+1$ .

c.

To determine

Find a system of inequalities.

Answer to Problem 46HP

  $y\ge x+3;y\le x+3$

Explanation of Solution

Given information:

The solution that lies only on a line.

Calculation:

The inequality that defines a region lying above and along the line $y=x+3$ is,

  $y\ge x+3$

The inequality that defines a region lying below and along the line $y=x+3$ is,

  $y\le x+3$

The only points that these inequalities share are those along the line $y=x+3$ are,

  $y\ge x+3;y\le x+3$

Hence, the inequality of a solution that lies only on a line is $y\ge x+3;y\le x+3$ .

d.

To determine

Find a system of inequalities.

Answer to Problem 46HP

  $y\le x;y\le -x;y\ge 0$.

Explanation of Solution

Given information:

The solution that lies on exactly one point.

Calculation:

The inequality that defines a region lying along and between the line $y=xandy=-x$ , which includes the origin and portions of quadrant $3and4$ is,

  $y\le x;y\le -x$

The inequality that defines a region lying below and along the $x-axis$ , including the origin is,

  $y\ge 0$

The only points that these inequalitieshave in common is the origin (this would not be true if strict inequalities were used). So, we get,

  $y\le x;y\le -x;y\ge 0$

Hence, the inequality of a solution that lies on exactly one point is $y\le x;y\le -x;y\ge 0$ .