Q41. - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line.

41. $\left|n\right|\ge n$

Step-by-Step Solution

Answer

Solution of inequality is all real number.

Solution set in number line is

Step 1 - Definition of absolute value inequality

For real numbers $a\text{ and }b$ with b>0

Absolute value inequality is $\left|a\right|<b$ when $-b<a\text{ and }a<b$, and

Absolute value inequality is $\left|a\right|>b$ when $a<-b\text{ or }a>b$ 

Step 2 - Write properties of inequality

Properties of inequalities are written in following table:

Step 3 - Simplify n &#8805; n

For $\left|a\right|>b$, b is always greater than 0

So let n is a positive number

From definition of absolute value inequality, given expression is simplified to 

$n\le -n\text{ or }n\ge n$  

Use division property of inequality, division by same positive number does not reverse the sign of inequality as shown below

$\begin{array}{c}n\le -n\text{ or }n\ge n\\ \frac{n}{n}\le -\frac{n}{n}\text{ or }\frac{n}{n}\ge \frac{n}{n}\\ 1\le -1\left(\text{False}\right)\text{ or 1}\ge 1\left(\text{True}\right)\end{array}$

Step 4 - Graph the solution set

As solution is all real number, so all points on number line are solution of given inequality.