Chapter 6, Problem 8SGR

To determine

To state: The given statement is true or false. If false replace underlined term to make a true sentence.

Answer to Problem 8SGR

The given statement is true.

Explanation of Solution

Given information:

The given statement:

When the graphs of the inequalities in a system of inequalities do not intersect, there are no solutions to the system.

Concept used:

The points of intersections of inequalities are the feasible solutions of system of inequalities.

Suppose $x+y\ge 2$ and $2x-y\le 2$ .

The graphs of these inequalities are shown in Figure-1.

The points that satisfy both the inequalities are the solutions of system of inequalities.

Consider points $\left(5,0\right)$ $\left(0,-5\right)$ $\left(0,5\right)$ $\left(0,-5\right)$ .

Point $\left(5,0\right)$ satisfies one inequalities

  $x+y\ge 2\Rightarrow$ $5+0>2$ . $2x-y\le 2\Rightarrow 2\cdot 5-0\cancel{\le }2$ .

Point $\left(0,-5\right)$ does not satisfy either of the inequalities

  $x+y\ge 2\Rightarrow$ $0-5\cancel{\ge }2$ . $2x-y\le 2\Rightarrow 2\cdot 0-\left(-5\right)\cancel{\le }2$ .

Point $\left(-5,0\right)$ satisfies one of the inequalities

  $x+y\ge 2\Rightarrow -5+0\cancel{\ge }2$ . $2x-y\le 2\Rightarrow 2\cdot \left(-5\right)-0<2$ .

Point $\left(0,5\right)$ satisfies both of the inequalities

  $x+y\ge 2\Rightarrow 0+5>2$ , $2x-y\le 2\Rightarrow 2\cdot \left(0\right)-5<2$ .

Thus, when the graphs of the inequalities in a system of inequalities do not intersect, there are no solutions to the system.

Hence, the given statement is true.

Conclusion:

The given statement is true.