Oct 23, 2017

$x = \left\{2 , \frac{16}{3}\right\}$

Explanation:

This equation can also be stated as

$\sqrt{{\left(x - 3\right)}^{2}} + \sqrt{{\left(2 x - 8\right)}^{2}} = 5$ and squaring both sides

${\left(x - 3\right)}^{2} + {\left(2 x - 8\right)}^{2} + 2 \sqrt{{\left(x - 3\right)}^{2}} \sqrt{{\left(2 x - 8\right)}^{2}} = 25$

Arranging and squaring again

$4 {\left(x - 3\right)}^{2} {\left(2 x - 8\right)}^{2} = {\left(25 - \left({\left(x - 3\right)}^{2} + {\left(2 x - 8\right)}^{2}\right)\right)}^{2}$ or

$4 {\left(x - 3\right)}^{2} {\left(2 x - 8\right)}^{2} - {\left(25 - \left({\left(x - 3\right)}^{2} + {\left(2 x - 8\right)}^{2}\right)\right)}^{2} = 0$ or

$3 \left(x - 10\right) \left(x - 2\right) x \left(3 x - 16\right) = 0$ and the potential solutions are

$x = \left\{0 , 2 , 10 , \frac{16}{3}\right\}$ and the feasible solutions are

$x = \left\{2 , \frac{16}{3}\right\}$ because they verify the original equation.

Oct 23, 2017

$x = \frac{16}{3} \mathmr{and} x = 2$

Explanation:

$| x - 3 | + | 2 x - 8 | = 5$
Start by adding color(red)(-|2x-8| to both sides.
$| x - 3 | \cancel{+ | 2 x - 8 |} \cancel{\textcolor{red}{- | 2 x - 8 |}} = 5 \textcolor{red}{- | 2 x - 8}$
$| x - 3 | = - | 2 x - 8 | + 5$
We know....
Either $x - 3 = - | 2 x - 8 | + 5$ or $x - 3 = - \left(- | 2 x - 8 | + 5\right)$
Let's start with part $1$
$x - 3 = - | 2 x - 8 | + 5$
Flip the equation to fill more comfortable
$- | 2 x - 8 | + 5 = x - 3$
We want to eliminate $5$ on the left side and transfer it to the other side, to do that, we need to add $\textcolor{red}{- 5}$ to both sides
$- | 2 x - 8 | \cancel{+ 5} \cancel{\textcolor{red}{- 5}} = x - 3 \textcolor{red}{- 5}$
$- | 2 x - 8 | = x - 8$
We need to cancel the negative sign in front of the absolute value. To do that, we need to divide both sides by $\textcolor{red}{- 1}$
$\frac{- | 2 x - 8 |}{\textcolor{red}{- 1}} = \frac{x - 8}{\textcolor{red}{- 1}}$
$| 2 x - 8 | = - x + 8$
We know either $2 x - 8 = x - 8 \mathmr{and} 2 x - 8 = - \left(- x + 8\right)$
$2 x - 8 = - x + 8$
Start by adding $\textcolor{red}{x}$ to both sides
$2 x - 8 + \textcolor{red}{x} = x + 8 + \textcolor{red}{x}$
$3 x - 8 = 8$
$3 x = 8 + 8$
$3 x = 16$
$x = \frac{16}{3}$
Solve for the second possibility
$2 x - 8 = - \left(- x + 8\right)$
$2 x - 8 = x - 8$
Combine like terms
$2 x - x = - 8 + 8$
$x = 0$ (doesn't work in original equation)

Part 2:
$x - 3 = - \left(- | 2 x - 8 | + 5\right)$ (Look at the first one to see what I'mtalking about)
Flip the equation
$| 2 x - 8 | - 5 = x - 3$ (transfer 5 on the right side)
$| 12 x - 8 | = x - 3 + 5$
$| 12 x - 8 | = x + 2$
We know either $2 x - 8 = x + 2 \mathmr{and} 2 x - 8 = - \left(x + 2\right)$
Let's start solving the first possibility
$2 x - 8 = x + 2$
Combine like terms
$2 x - x = 2 + 8$
$x = 10$
Solve the second possibility
$2 x - 8 = - \left(x + 2\right)$
$2 x - 8 = - x - 2$
Combine like terms
$2 x + x = - 2 + 8$
$3 x = 6$
$x = \frac{6}{3}$
$= 2$ (Works in original equation)

Thus,
The final answer are $x = \frac{16}{3} \mathmr{and} x = 2$