Chapter 4.9, Problem 12WE

To determine

To find and graph the solutionset of inequality.

Answer to Problem 12WE

The solution set of given inequality is $\left\{r:4\ge r\ge -2\right\}$

Explanation of Solution

Given information:

The given inequality is $r^2\le 2\left(r+4\right)$

Calculation:

First write the inequality

  $\begin{array}{l}{r}^2\le 2\left(r+4\right)\\ r^2-2\left(r+4\right)\le 0\left[\text{make one side 0}\right]\\ r^2-2r-8\le 0\\ \left(r-4\right)\left(r+2\right)\le 0\left[\text{factor of the polynomial}\right]\end{array}$

The product is negative or 0 if and only if both the factors have opposite sign or at least one factor must be zero.

Case 1:

  $\begin{array}{l}r-4\ge 0\text{ and }r+2\le 0\\ r\ge 4\text{ and }r\le -2\end{array}$

No solution

Case 2:

  $\begin{array}{l}r-4\le 0\text{ and }r+2\ge 0\\ r\le 4\text{ and }r\ge -2\end{array}$

The solution set of this conjunction is $\left\{r:4\ge r\ge -2\right\}$

Therefore,

The solution set of given inequality is $\left\{r:4\ge r\ge -2\right\}$

The graph of the solution set is given below,