Q42. - Algebra 2

Question

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

$\left|n\right|\le n$

Step-by-Step Solution

Answer

Solution of inequality is all non-negative 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 - 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 

$-n\le n\text{ and }n\le n$

Since, $-n\le n$ is true only when n is positive real number and $n\le n$ is true for all real number.

Solution is “Non-negative real number” and “All real number”

The solution of compound inequality containing and is the intersection of the solution set of the two inequalities.

So solution is non-negative real number.

Step 3 - Graph the solution set

As solution is all non-negative real number, so all points on number line start from 0 to $\infty$ are solution of given inequality.