Question

Solve$\left|3x-5\right|-1=6$

Short answer

The required solutions are$x_1=-\frac{2}{3}\hspace{0.17em},x_2=4$.

Step-by-step solution

Step 1. Given information. 

We have:

$\left|3x-5\right|-1=6$

Step 2. Isolate the absolute value expression 

We have:

$\left|3x-5\right|-1=6$

Add $1$to both sides:

$$\begin{gathered} \left|3x-5\right|-1+1=6+1 \\ \left|3x-5\right|=7 \end{gathered}$$

Step 4. Solve each equation 

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

First, solve $3x-5=-7:$

Add $5$to both sides:

$$\begin{gathered} 3x-5+5=-7+5 \\ 3x=-2 \\ \frac{3x}{3}=\frac{-2}{3} \\ x=-\frac{2}{3} \end{gathered}$$

Now, solve the second equation:

$3x-5=7$

Add $5$to both sides:

$$\begin{gathered} 3x-5+5=7+5 \\ 3x=12 \\ \frac{3x}{3}=\frac{12}{3} \\ x=4 \end{gathered}$$

Step 4. Check the each solution 

Substitute $-\frac{2}{3}$and $4$ into the original equation::

$\left|3x-5\right|-1=6$

At $x=-\frac{2}{3}:$

$$\begin{gathered} \left|3\left(-\frac{2}{3}\right)-5\right|-1=6 \\ 7-1=6 \\ 6=6 \end{gathered}$$

Hence, verified.

At $x=4:$

$$\begin{gathered} \left|3\left(4\right)-5\right|-1=6 \\ 7-1=6 \\ 6=6 \end{gathered}$$

Hence, verified.