Question

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

$\left|2m\right|\ge 8$

Short answer

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

Solution set in number line is

Step-by-step solution

- 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 $a<-b\text{or}a>b$

- Write properties of inequality

Properties of inequalities are written in following table:

- Simplify 2m≥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

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

- 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.