Question

How do you solve `3/4abs(y+2)=6`?

Answer 1

See a solution process below:

Explanation:

First, multiply each side of the equation by `color(red)(4)/color(blue)(3)` to isolate the absolute value function while keeping the equation balanced:

`color(red)(4)/color(blue)(3) xx 3/4abs(y + 2) = color(red)(4)/color(blue)(3) xx 6`

`12/12abs(y + 2) = 24/color(blue)(3)`

`1abs(y + 2) = 8`

`abs(y + 2) = 8`

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:

`y + 2 = -8`

`y + 2 - color(red)(2) = -8 - color(red)(2)`

`y + 0 = -10`

`y = -10`

Solution 2:

`y + 2 = 8`

`y + 2 - color(red)(2) = 8 - color(red)(2)`

`y + 0 = 6`

`y = 6`

The Solution Set Is:

`x = {-10, 6}`

Answer 2

`y= -10, 6`

Explanation:

Because you don't know if y is positive or negative, `|y+2|` equals both `(y+2)` and `-(y+2)`

Start by dividing both sides by `3/4`:

`|y+2|=6(4/3)`

`|y+2|=8`

Then set up two equations and solve:

`y+2=8`

`y=6`

and

`-(y+2)=8`

`-y-2=8`

`-y=10`

Multiply both sides by `-1`

`-y xx -1 = 10 xx -1`

`y = -10`