Chapter 4.9, Problem 12OE

To determine

To justify the statements

Explanation of Solution

Given information:

The given statement is that the solution set of $\left(x-2\right)\left(x^2+1\right)<0$ is $\left\{x:x<2\right\}$ .

Proof:

First write the inequality

  $\left(x-2\right)\left(x^2+1\right)<0$

Since the factor $\left(x^2+1\right)$ is always positive for any real number because square of any real number is a non-negative real number so to get a value less than 0, $\left(x-2\right)$ must be less than 0

  $\begin{array}{l}\left(x-2\right)<0\\ x<2\left[\text{add 2 to each side}\right]\end{array}$

To satisfy the inequality x must be less than 2

Hence, the solution set of $\left(x-2\right)\left(x^2+1\right)<0$ is $\left\{x:x<2\right\}$ .