Question

Solve$\left|x\right|>2$. Graph the solution and write the solution in interval notation.

Short answer

The solution of the inequality is $x<-2,x>2$.

Interval notation: $\left(-\infty \hspace{0.17em},\hspace{0.17em}-2\right)\cup \left(2,\hspace{0.17em}\infty \hspace{0.17em}\right)$

The graph on the number line is:

Step-by-step solution

Step 1. Write the equivalent inequality. 

We have:

$\left|x\right|>2$

Apply absolute rule: $\mathrm{If}\hspace{0.17em}\left|u\right|\hspace{0.17em}>\hspace{0.17em}a,\hspace{0.17em}a>0\hspace{0.17em}\mathrm{then}\hspace{0.17em}u\hspace{0.17em}<\hspace{0.17em}-a\hspace{0.17em}\mathrm{or}\hspace{0.17em}u\hspace{0.17em}>\hspace{0.17em}a$

$x<-2\mathrm{or}\hspace{0.17em}x>2$

Step 2. Graph the solution. 

Plot the point $-2$on the number line with an open circle since it is not included in the inequality. Plot the point $2$on the number line with open circle since it is not included in the inequality.

Draw on the number line:

Step 3. Check the solution.

To verify, check the value in each section of the number line showing the solution. Choose the number such as $-4,0,$and$7.$

$$\begin{gathered} \left|x\right|>\hspace{0.17em}2 \\ \left|-4\right|>\hspace{0.17em}2 \\ 4>2 \end{gathered}$$

True.

$$\begin{gathered} \left|x\right|>\hspace{0.17em}2 \\ \left|0\right|>\hspace{0.17em}2 \\ 0>2 \end{gathered}$$

False.

$$\begin{gathered} \left|x\right|>\hspace{0.17em}2 \\ \left|7\right|>\hspace{0.17em}2 \\ 7>2 \end{gathered}$$

True.