Chapter 4.9, Problem 16OE

To determine

To write the solution set of the inequality.

Answer to Problem 16OE

The solution set of the inequality is $\left\{x:4\ge x\ge 0\right\}$

Explanation of Solution

Given information:

The given inequality is $x\left(x-4\right)\le 0$

Calculation:

First write the inequality

  $x\left(x-4\right)\le 0$

The product is negative or equals to zero if and only if both factors have opposite sign or at least one of them equals to zero.

Case 1: when x is positive and $\left(x-4\right)$ is negative or at least one of them equals to zero.

  $\begin{array}{l}x\ge 0\text{ and }\left(x-4\right)\le 0\\ x\le 4\left[\text{ add }\left(4\right)\text{to each side}\right]\end{array}$

The solution set of this conjunction is $\left\{x:4\ge x\ge 0\right\}$

Case 2: when x is negative and $\left(x-4\right)$ is also negative

The solution set of this conjunction is $\varphi$ .

Therefore,

The solution set of the inequality is $\left\{x:4\ge x\ge 0\right\}$