Chapter 4.9, Problem 10WE

To determine

To find and graph the solutionset of inequality.

Answer to Problem 10WE

The solution set of given inequality is $\left\{z:z>3\text{ or z}<-2\right\}$

Explanation of Solution

Given information:

The given inequality is $z^2-z-6>0$

Calculation:

First write the inequality

  $\begin{array}{l}{z}^2-z-6>0\\ \left(z+2\right)\left(z-3\right)\left[\text{factor of the polynomial}\right]\end{array}$

The product is positive if and only if both the factors have same sign.

Case 1: Both factors positive

  $\begin{array}{l}z+2>0\text{ and z}-3>0\\ z>-2\text{ and z}>3\end{array}$

The solution set of this conjunction is $\left\{z:z>3\right\}$

Case 2: Both factors negative

  $\begin{array}{l}z+2<0\text{ and z}-3<0\\ z<-2\text{ and z}<3\end{array}$

The solution set of this conjunction is $\left\{z:z<-2\right\}$

Therefore,

The solution set of given inequality is $\left\{z:z>3\text{ or z}<-2\right\}$

The graph of the solution set is given below