Q34. - Algebra 2

Question

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

 $\left|2m\right|\ge 8$

Step-by-Step Solution

Answer

Solution of inequality is $m\le -4\text{ or }m\ge 4$ 

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 2 m &#8805; 8

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

$2m\le -8\text{ or }2m\ge 8$  

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

$\begin{array}{c}\frac{2m}{2}\le -\frac{8}{2}\text{ or }\frac{2m}{2}\ge \frac{8}{2}\\ m\le -4\text{ or }m\ge 4\end{array}$

Step 3 - Graph the solution set

For $m\le -4\text{ or }m\ge 4$, number line starts from $-4$ to $-\infty$ and from 4 to $\infty$ 

Dot on $-4\text{ and 4}$ shows these points are included in the solution.