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

First, add `color(red)(9)` to each side of the equation to isolate the absolute value term while keeping the equation balanced:

`7abs(10v - 2) - 9 + color(red)(9) = 5 + color(red)(9)`

`7abs(10v - 2) - 0 = 14`

`7abs(10v - 2) = 14`

Next, divide each side of the equation by `color(red)(7)` to isolate the absolute value function while keeping the equation balanced:

`(7abs(10v - 2))/color(red)(7) = 14/color(red)(7)`

`(color(red)(cancel(color(black)(7)))abs(10v - 2))/cancel(color(red)(7)) = 2`

`abs(10v - 2) = 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:

`10v - 2 = -2`

`10v - 2 + color(red)(2) = -2 + color(red)(2)`

`10v - 0 = 0`

`10v = 0`

`(10v)/color(red)(10) = 0/color(red)(10)`

`(color(red)(cancel(color(black)(10)))v)/cancel(color(red)(10)) = 0`

`v = 0`

Solution 2:

`10v - 2 = 2`

`10v - 2 + color(red)(2) = 2 + color(red)(2)`

`10v - 0 = 4`

`10v = 4`

`(10v)/color(red)(10) = 4/color(red)(10)`

`(color(red)(cancel(color(black)(10)))v)/cancel(color(red)(10)) = 2/5`

`v = 2/5`

The Solution Set Is: `v = {0, 2/5}`