# How do you solve 10-9|7x-5|=-12?

Jan 28, 2017

See the entire solution process below:

#### Explanation:

First, subtract $\textcolor{red}{10}$ from each side of the equation to isolate the absolute value term while keeping the equation balanced:

$10 - 9 \left\mid 7 x - 5 \right\mid - \textcolor{red}{10} = - 12 - \textcolor{red}{10}$

$10 - \textcolor{red}{10} - 9 \left\mid 7 x - 5 \right\mid = - 22$

$0 - 9 \left\mid 7 x - 5 \right\mid = - 22$

$- 9 \left\mid 7 x - 5 \right\mid = - 22$

Next, divide each side of the equation by $\textcolor{red}{- 9}$ to be able to solve for the absolute value term while keeping the equation balanced:

$\frac{- 9 \left\mid 7 x - 5 \right\mid}{\textcolor{red}{- 9}} = \frac{- 22}{\textcolor{red}{- 9}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 9}}} \left\mid 7 x - 5 \right\mid}{\cancel{\textcolor{red}{- 9}}} = \frac{22}{9}$

$\left\mid 7 x - 5 \right\mid = \frac{22}{9}$

The absolute value function transforms any negative or positive term into its positive form. Therefore the term in side the absolute value must be solve for the positive and negative form of what it is equated to.

$\textcolor{red}{S o l u t i o n 1}$

$7 x - 5 = \frac{22}{9}$

$7 x - 5 + \textcolor{red}{5} = \frac{22}{9} + \textcolor{red}{5}$

$7 x - 0 = \frac{22}{9} + \left(\frac{9}{9} \times \textcolor{red}{5}\right)$

$7 x = \frac{22}{9} + \frac{45}{9}$

$7 x = \frac{67}{9}$

$7 x \times \frac{1}{\textcolor{red}{7}} = \frac{67}{9} \times \frac{1}{\textcolor{red}{7}}$

$\textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} x \times \frac{1}{\cancel{\textcolor{red}{7}}} = \frac{67 \times 1}{9 \times 7}$

$x = \frac{67}{63}$

$\textcolor{red}{S o l u t i o n 2}$

$7 x - 5 = - \frac{22}{9}$

$7 x - 5 + \textcolor{red}{5} = - \frac{22}{9} + \textcolor{red}{5}$

$7 x - 0 = - \frac{22}{9} + \left(\frac{9}{9} \times \textcolor{red}{5}\right)$

$7 x = - \frac{22}{9} + \frac{45}{9}$

$7 x = \frac{23}{9}$

$7 x \times \frac{1}{\textcolor{red}{7}} = \frac{23}{9} \times \frac{1}{\textcolor{red}{7}}$

$\textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} x \times \frac{1}{\cancel{\textcolor{red}{7}}} = \frac{23 \times 1}{9 \times 7}$

$x = \frac{23}{63}$

The solutions are: $x = \frac{67}{63}$ and $x = \frac{23}{63}$