Question

Solve$\left|4x-3\right|-5=2$

Short answer

The required solutions are$x_1=-1,x_2=\frac{5}{2}$.

Step-by-step solution

Step 1. Given information.  

We have:

$\left|4x-3\right|-5=2$

Step 2. Isolate the absolute expression. 

We have:

$\left|4x-3\right|-5=2$

Add $5$to both sides:

$$\begin{gathered} \left|4x-3\right|-5+5=2+5 \\ \left|4x-3\right|=7 \end{gathered}$$

Step 3. Write the equivalent equations. 

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

$4x-3=-7\mathrm{or}\hspace{0.17em}4x-3=7$

Step 4. Solve the each equations. 

Start with first equation:

$4x-3=-7$

Add $3$to both sides:

$$\begin{gathered} 4x-3+3=-7+3 \\ 4x=-4 \\ \frac{4x}{4}=\frac{-4}{4} \\ x=-1 \end{gathered}$$

Now, solve the second equation:

$4x-3=7$

Add $3$to both sides:

$$\begin{gathered} 4x-3+3=7+3 \\ 4x=10 \\ \frac{4x}{4}=\frac{10}{4} \\ x=\frac{5}{2} \end{gathered}$$

Step 5. Check the each solution. 

Substitute $-1$and $\frac{5}{2}$into the original equation:

$\left|4x-3\right|-5=2$

At $x=-1$:

$$\begin{gathered} \left|4\left(-1\right)-3\right|-5=2 \\ 7-5=2 \\ 2=2 \end{gathered}$$

Hence, verified.

At $x=\frac{5}{2}$:

$$\begin{gathered} \left|4\left(\frac{5}{2}\right)-3\right|-5=2 \\ 7-5=2 \\ 2=2 \end{gathered}$$

Hence, verified.