Chapter 1.2, Problem 2QR

To determine

To find:The solution of the inequality $x\left(x-2\right)>0$ .

Answer to Problem 2QR

The solution of the inequality $x\left(x-2\right)>0$ is the interval $\left(-\infty ,0\right)\cup \left(2,\infty \right)$ .

Explanation of Solution

Given information:The given inequality is $x\left(x-2\right)>0$ .

Calculation:

There are two casesfor the inequality $x\left(x-2\right)>0$ .

Case 1: Both $x>0$ and $x-2>0$ .

For $x>0$ , the solution of the inequality is the interval $\left(0,\infty \right)$ .

Case 2: Both $x<0$ and $x-2<0$ .

For $x-2>0$ ;

Simplify the inequality.

  $\begin{array}{c}x-2>0\\ x>2\end{array}$

The solution of the inequality $x-2>0$ is the interval $\left(2,\infty \right)$ .The common interval of both the solution intervals $\left(0,\infty \right)$ and $\left(2,\infty \right)$ is $\left(2,\infty \right)$ .

For $x<0$ ;

The solution of the inequality $x<0$ is the interval $\left(-\infty ,0\right)$ .

Now substitute the factor $x-2<0$ .

Simplify the inequality.

  $\begin{array}{c}x-2<0\\ x<2\end{array}$

The solution of the inequality $x<2$ is the interval $\left(-\infty ,2\right)$ .The common interval of both the solution intervals $\left(-\infty ,0\right)$ and $\left(-\infty ,2\right)$ is $\left(-\infty ,0\right)$ .

So, the solution of the inequality $x\left(x-2\right)>0$ is the union of the intervals $\left(-\infty ,0\right)$ and $\left(2,\infty \right)$ .

Therefore, the solution of the inequality $x\left(x-2\right)>0$ is the interval $\left(-\infty ,0\right)\cup \left(2,\infty \right)$ .