Chapter 4.9, Problem 11WE

To determine

To find and graph the solutionset of inequality.

Answer to Problem 11WE

The solution set of given inequality is $\left\{t:t>6\text{ or t}<3\right\}$

Explanation of Solution

Given information:

The given inequality is $t^2>9\left(t-2\right)$

Calculation:

First write the inequality

  $\begin{array}{l}{t}^2>9\left(t-2\right)\\ t^2-9\left(t-2\right)>0\left[\text{make one side 0}\right]\\ t^2-9t+18>0\\ \left(t-6\right)\left(t-3\right)>0\left[\text{factor of the polynomial}\right]\end{array}$

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

Case 1: Both factors positive

  $\begin{array}{l}t-6>0\text{ and }t-3>0\\ t>6\text{ and }t>3\end{array}$

The solution set of this conjunction is $\left\{t:t>6\right\}$

Case 2: Both factors negative

  $\begin{array}{l}t-6<0\text{ and }t-3<0\\ t<6\text{ and }t<3\end{array}$

The solution set of this conjunction is $\left\{t:t<3\right\}$

Therefore,

The solution set of given inequality is $\left\{t:t>6\text{ or t}<3\right\}$

The graph of the solution set is given below