# How do you solve the equation 2abs(5x+1)-3=0?

Aug 23, 2017

Given: $2 | 5 x + 1 | - 3 = 0$

$2 | 5 x + 1 | = 3$

Divide both sides by 2

$| 5 x + 1 | = \frac{3}{2}$

Separate into two equations without the absolute value function, one with the right side positive and the other with the right side negative:

$5 x + 1 = \frac{3}{2}$ and $5 x + 1 = - \frac{3}{2}$

Subtract one from both sides of both equations:

$5 x = \frac{1}{2}$ and $5 x = - \frac{5}{2}$

Divide both sides of both equations by 5:

$x = \frac{1}{10}$ and $x = - \frac{1}{2}$

Check:

$2 | 5 \left(\frac{1}{10}\right) + 1 | - 3 = 0$ and $2 | 5 \left(- \frac{1}{2}\right) + 1 | - 3 = 0$

$2 | \frac{3}{2} | - 3 = 0$ and $2 | - \frac{3}{2} | - 3 = 0$

$3 - 3 = 0$ and $3 - 3 = 0$

Both values check.

Aug 23, 2017

$X = \frac{1}{10} \mathmr{and} x = - \frac{1}{2}$

#### Explanation:

$2 | 5 x + 1 | - 3 = 0$
$2 | 5 x + 1 | = 0 + 3$
$2 | 5 x + 1 | = 3$
$| 5 x + 1 | = \frac{3}{2}$
We know either
$5 x + 1 = \frac{3}{2} \mathmr{and} 5 x + 1 = - \frac{3}{2}$
Let's solve the first one
$5 x + 1 = \frac{3}{2}$
$5 x = \frac{3}{2} - 1$
$5 x = \frac{1}{2}$
Divide both sides by 5
$5 \frac{x}{5} = \frac{1}{2} / 5$
$x = \frac{1}{10}$
Solve the second one
$5 x + 1 = - \frac{3}{2}$
$5 x = - \frac{3}{2} - 1$
$5 x = - \frac{5}{2}$
Divide both sides by 5
$5 \frac{x}{5} = - \frac{5}{2} / 5$
$x = - \frac{1}{2}$

Therefore,
$x = \frac{1}{10} \mathmr{and} x = - \frac{1}{2}$

Aug 23, 2017

$x = - \frac{1}{2} \text{ or } x = \frac{1}{10}$

#### Explanation:

$\text{isolate the absolute value}$

$\text{add 3 to both sides}$

$2 | 5 x + 1 | \cancel{- 3} \cancel{+ 3} = 0 + 3$

$\Rightarrow 2 | 5 x + 1 | = 3$

$\text{divide both sides by 2}$

$\frac{\cancel{2}}{\cancel{2}} | 5 x + 1 | = \frac{3}{2}$

$\Rightarrow | 5 x + 1 | = \frac{3}{2}$

$\text{the expression inside the absolute value can be}$
$\text{positive or negative}$

$\textcolor{b l u e}{\text{Solution 1}}$

$5 x + 1 = \frac{3}{2}$

$\text{subtract 1 from both sides}$

$\Rightarrow 5 x = \frac{1}{2}$

$\text{dividing both sides by 5 gives}$

$\Rightarrow \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{x = \frac{1}{10}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\textcolor{b l u e}{\text{Solution 2}}$

$- \left(5 x + 1\right) = \frac{3}{2}$

$\Rightarrow - 5 x - 1 = \frac{3}{2}$

$\text{add 1 to both sides}$

$\Rightarrow - 5 x = \frac{5}{2}$

$\text{divide both sides by - 5}$

$\Rightarrow \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{x = - \frac{1}{2}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$