Chapter 6.6, Problem 29PPE

a.

To determine

To determine the graph of the following system.

Explanation of Solution

Given:

  $y>3x+3$

  $y\le 3x-5$

The inequality expressions are:

  $y>3x+3$

  $y\le 3x-5$

Graph:

  

Interpretation:

Therefore, this is the the graph of the following system.

b.

To determine

To determine whether the system of equation intersect or not.

Answer to Problem 29PPE

No

Explanation of Solution

Given:

  $y=3x-5$

  $y=3x+3$

If the system of equation intersects then it should satisfy each other.

  $y=3x-5\to \text{(i)}$

  $y=3x+3\to \text{(ii)}$

Putting the value of (i) in equation (ii),

  $3x-5=3x+3$$\left[\because y=3x-5\right]$

  $-5\ne 3$

So it does not satisfy the equation.

Conclusion:

Therefore, the system of equation $y=3x-5$ and $y=3x+3$ does not intersect.

c.

To determine

To determine whether the shaded region in the graph from (a) overlap or not.

Answer to Problem 29PPE

No

Explanation of Solution

Given:

The shaded region in the graph from (a)

From the graph in (a) the representation is,

Graph:

  

And from (b) the system of equation $y>3x+3$ and $y\le 3x-5$ does not intersect.

Moreover, the shaded portion for $y>3x+3$ is above and the shaded portion for $y\le 3x-5$ is below.

There is no overlap of the two graph as in the middle there is empty portion that does not share any property with $y>3x+3$ and $y\le 3x-5$ .

Conclusion:

Therefore, the shaded region in the graph from (a) does not overlap.

c.

To determine

To determine whether the system of inequality have any solution.

Answer to Problem 29PPE

No

Explanation of Solution

Given:

The system of inequality

  $y>3x+3$ and $y\le 3x-5$

From the graph in (a) the representation is,

Graph:

  

And from (b) the system of equation $y>3x+3$ and $y\le 3x-5$ does not intersect.

Moreover, the shaded portion for $y>3x+3$ is above and the shaded portion for $y\le 3x-5$ is below.

There is no overlap of the two graph as in the middle there is empty portion that does not share any property with $y>3x+3$ and $y\le 3x-5$ .

Thus, there is no solution of the system $y>3x+3$ and $y\le 3x-5$ as it does not intersect or overlap each other. The system $y>3x+3$ and $y\le 3x-5$ run up to infinity without any intersection.

Conclusion:

Therefore, the system of inequality does not have any solution.