Chapter 6.8, Problem 39HP

To determine

To state: whether the given statement is sometimes, always or never true

Answer to Problem 39HP

The solution will be at all the points such that $y\le 1$

Systems with parallel boundary has no solutions

It is also possible that the solution could be one of the regions determined by the parallel boundaries

Explanation of Solution

Given:

Systems of inequalities with parallel boundaries have no solutions

Calculation:

The system of two equations would have parallel boundaries and have no solution if there is no any common region between two lines.

Consider the below two inequality equations:

  $\begin{array}{l}2x-y>3\\ 2x-y<-1\end{array}$

The graph will be

  

Again the system given be the inequality equations, $y\le 3andy\le 1$ also have parallel boundaries but the solution will contain an infinite number of points as one of the shaded regions completely overlaps the other.

  

The solution would be at all the points such that $y\le 1$.

Hence it is sometimes true that systems with parallel boundaries have no solution.

It is also possible that the solution could be one of the regions determined by the parallel boundaries

Conclusion:

The solution will be at all the points such that $y\le 1$

Systems with parallel boundary has no solutions

It is also possible that the solution could be one of the regions determined by the parallel boundaries