Chapter 4.9, Problem 30WE

To determine

To find and graph the solution of given inequality

Answer to Problem 30WE

The solution of the given inequality is $\left\{x:-2\le x\le 1\text{ or }x=2\right\}$ and the graph is represented.

Explanation of Solution

Given information:

The inequality is $\left(x^2+x-2\right)\left(x^2-4x+4\right)\le 0$

The solution of the given inequality can be obtained as,

  $\begin{array}{l}\left(x^2+x-2\right)\left(x^2-4x+4\right)\le 0\left[\text{Original equation}\right]\\ \left(x+2\right)\left(x-1\right){\left(x-2\right)}^2\le 0\left[\text{Factor}\right]\end{array}$

Note that ${\left(x-2\right)}^2\le 0$ is true for all value of $x$ , hence the solution of ${\left(x-2\right)}^2\le 0$ are same as $\left(x+2\right)\left(x-1\right){\left(x-2\right)}^2\le 0$

The sign of the product $\left(x+2\right)\left(x-1\right){\left(x-2\right)}^2\le 0$ is determined by the number of negative factors. When exactly one or all the three factors are negative then the product is negative. When no factors or exactly two factors are negative, the product is positive.

Since the inequality $\le$ contains the ‘=’ sign it can be concluded that the number that has found can be solved as a part of the equation

  

Therefore, the solution of the given inequality is $\left\{x:-2\le x\le 1\text{ or }x=2\right\}$

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