How do you solve 7abs(10v-2)-9=5?

Mar 15, 2018

See a solution process below:

Explanation:

First, add $\textcolor{red}{9}$ to each side of the equation to isolate the absolute value term while keeping the equation balanced:

$7 \left\mid 10 v - 2 \right\mid - 9 + \textcolor{red}{9} = 5 + \textcolor{red}{9}$

$7 \left\mid 10 v - 2 \right\mid - 0 = 14$

$7 \left\mid 10 v - 2 \right\mid = 14$

Next, divide each side of the equation by $\textcolor{red}{7}$ to isolate the absolute value function while keeping the equation balanced:

$\frac{7 \left\mid 10 v - 2 \right\mid}{\textcolor{red}{7}} = \frac{14}{\textcolor{red}{7}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} \left\mid 10 v - 2 \right\mid}{\cancel{\textcolor{red}{7}}} = 2$

$\left\mid 10 v - 2 \right\mid = 2$

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

Solution 1:

$10 v - 2 = - 2$

$10 v - 2 + \textcolor{red}{2} = - 2 + \textcolor{red}{2}$

$10 v - 0 = 0$

$10 v = 0$

$\frac{10 v}{\textcolor{red}{10}} = \frac{0}{\textcolor{red}{10}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{10}}} v}{\cancel{\textcolor{red}{10}}} = 0$

$v = 0$

Solution 2:

$10 v - 2 = 2$

$10 v - 2 + \textcolor{red}{2} = 2 + \textcolor{red}{2}$

$10 v - 0 = 4$

$10 v = 4$

$\frac{10 v}{\textcolor{red}{10}} = \frac{4}{\textcolor{red}{10}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{10}}} v}{\cancel{\textcolor{red}{10}}} = \frac{2}{5}$

$v = \frac{2}{5}$

The Solution Set Is: $v = \left\{0 , \frac{2}{5}\right\}$