Question

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

$\left|b-4\right|>6$

Short answer

Solution of inequality is $b<-2\text{or}b>10$

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 b-4>6

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

$b-4<-6\text{or}b-4>6$

Use addition property of inequality, add with same number does not reverse the sign of inequality as shown below

$\begin{array}{c}b-4+4<-6+4\text{or}b-4+4>6+4\\ b<-2\text{or}b>10\end{array}$

- Graph the solution set

For $b<-2\text{or}b>10$, number line starts from $-2$ to $-\infty$ and from 10 to $\infty$

Open circle on $-2\text{and 10}$ shows these points are not included in the solution.