Question

Solve$3\left|x-4\right|-4=8$

Short answer

The required solutions are$x_1=0,x_2=8$.

Step-by-step solution

Step 1. Given information.

We have:

$3\left|x-4\right|-4=8$

Step 2. Isolate the absolute expression.  

We have:

$3\left|x-4\right|-4=8$

Add $4$to both sides:

$$\begin{gathered} 3\left|x-4\right|-4+4=8+4 \\ 3\left|x-4\right|=12 \\ \frac{3\left|x-4\right|}{3}=\frac{12}{3} \\ \left|x-4\right|=4 \end{gathered}$$

Step 3. Write the equivalent equations.  

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

$x-4=-4\mathrm{or}\hspace{0.17em}x-4=4$

Step 4. Solve the each equations.  

Start with first equation:

$x-4=-4$

Add $4$to both sides:

$$\begin{gathered} x-4+4=-4+4 \\ x=0 \end{gathered}$$

Now, solve $x-4=4:$

$$\begin{gathered} x-4+4=4+4 \\ x=8 \end{gathered}$$

Step 5. Check the each solution. 

Substitute $0$and $8$into the original equation:

$3\left|x-4\right|-4=8$

At $x=0:$

$$\begin{gathered} 3\left|0-4\right|-4=8 \\ 12-4=8 \\ 8=8 \end{gathered}$$

Hence, verified

At $x=8:$

$$\begin{gathered} 3\left|8-4\right|-4=8 \\ 12-4=8 \\ 8=8 \end{gathered}$$

Hence, verified.