Chapter 4.9, Problem 22WE

To determine

To find and graph the solution of given inequality

Answer to Problem 22WE

The solution of the given inequality is $\left\{z:z>0\text{ or }z<-2\text{ and }z>-5\right\}$ and the graph is represented

Explanation of Solution

Given information:

The inequality is $z^3+7z^2+10z>0$

The solution of the given expression can be obtained as,

  $\begin{array}{l}{z}^3+7z^2+10z>0\left[\text{Original equation}\right]\\ z\left(z^2+7z+10\right)>0\\ z\left(z+2\right)\left(z+5\right)>0\left[\text{Factor the polynomial}\right]\end{array}$

To draw a sign graph, draw a number line for each factor and label the part where the factor is positive, where it is zero, and where it is negative

For $z$

  

For $\left(z+2\right)$

  

For $\left(z+5\right)$

  

The sign of the factor $z\left(z+2\right)\left(z+5\right)$ is determined by the number of negative factors. When exactly one or exactly three factors are negative, the product is negative. When no factors or exactly two factors are negative, the product is positive.

For $z\left(z+2\right)\left(z+5\right)$

  

Therefore, the solution of the given inequality is $\left\{z:z>0\text{ or }z<-2\text{ and }z>-5\right\}$

Now, the graph of the solution set can be represented as,