Chapter 5.7, Problem 32E

To determine

To describe: the intersection of the half planes of the system shown.

Answer to Problem 32E

The system has no solution.

Explanation of Solution

Given information:

Given two inequalities

  $\begin{array}{l}x-y\le 4\\ x-y\ge 4\end{array}$

Calculation:

Rewrite the inequality 1 in slope intercept form as shown below

  $x-y\le 4$

Subtract $x$ from both sides of equation.

  $\begin{array}{c}x-x-y\le 4-x\\ -y\le 4-x\end{array}$

Multiply by negative sign from both sides of equation and reverse the inequality sign as shown below.

  $y\ge -x+4$

The shaded region is the upper half of the solid boundary line $y=-x+4$

Rewrite the inequality 2 in slope intercept form as shown below.

  $x-y\ge 4$

Subtract from both sides of the equation.

  $\begin{array}{c}x-x-y\ge 4-x\\ -y\ge 4-x\end{array}$

Multiply by negative sign from both sides of equation and reverse the inequality sign as shown below.

  $-y\le x-4$

Therefore, there is no common shaded region. Since, the lines are parallel and the half planes do not intersect.

Hence, the system has no solution.