Chapter 5.7, Problem 32E

To determine

To find: the intersection of the half plane of the system shown.

Answer to Problem 32E

The system has no solution.

Explanation of Solution

Given:

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

Concept used:

The difference is that a solution to the inequality is not the drawn line but area of the coordinate plane that satisfy the inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

Parallel line never intersects each other.

Calculation:

Consider the system of linear inequalities.

Inequality $1$ :

  $x-y\le 4.$

Inequality $2$ :

  $x-y\ge 4.$

Step $1$ : rewrite the inequality $1$ in the slope intercept form as show below.

  $x-y\le 4.$

Subtract $x$ from both the sides of the equation.

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

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 $y=-x+4$ .

Step $2$ : rewrite the inequality $2$ in slope intercept form as below:

  $x-y\le 4.$

Subtract from the both sides of the equation.

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

Multiply by negative sign from both the 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.