Question

How do you solve `2+8|10v-10|=82`?

Answer 1

v = 0 and v = 2.

Explanation:

I10v - 10I = `(82 - 2)/8 = 10`
Separate into 2 cases:
1. (10v - 10) = 10 --> 10v = 10 + 10 = 20 --> v = 2
2. (10v - 10) = -10 --> 10v = 0 --> v = 0

Check:
When v = 2 --> (20 - 10) = 10. OK
When v = 0 --> (0 -10) = -10 OK

Answer 2

`color(green)(v= 0" ; "v=+2)`

Explanation:

Given:`" "2+8|10v-10|=82`

Subtract 2 from both sides

`8xx|10v-10|=80`

Factor 10 out of the 'Absolute Value'

`8xx10|v-1|=80`

Divide both sides by 80

`|v-1|=1`

'~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Suppose "v =0" then")" "color(magenta)(" (corrected from "v < 0")"`

`v-1=-1" "`

`color(blue)(v=0 " so that " " | 0-1" "|" "=" "| -1|" " =" " + 1)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Suppose "v>0" then")" "color(magenta)(" ( corrected from "v>=0")")`

Then `+v-1=+1`

`color(blue)(v=2" So "|2-1|=|1| = +1)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(magenta)("Addition")`

`color(blue)("Investigate "x < 0)`

In this case `v-1 <-1 " so " |v-1| > 1`

Example: Suppose `v= -1" then we have" -1-1=-2`

and `|-2|!=+1`

`color(blue)("So "v " not" < 0)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Final solution")`

`color(green)(v= 0" ; "v=+2)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
'~~~~~~`color(red)(" Teaching bit about negative numbers")`~~~~~~~
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("A way of thinking about this to help understanding")`

`color(brown)("Opinion is divided on this concept. So just think of it as one of several ideas")`

Technically there is a very big difference between subtract/minus and the state of being negative.

Consider the value of negative 1
Although not commonly used, negative 1 should be written as `color(white)(.)^(-) 1`

The thing is: the vast majority of people write this as `-1` because this is how they have been taught in lower school. The problem is that the minus sign means just that: remove or subtract.

It is an action! If they wish to do this they should write: `0-1`. After this 'action' has been applied you end up at the state of negative 1. That is`" "color(white)(.)^("-") 1`

In the same way you have `color(white)(.)^(+)1`

In fact the 'minus' can be `color(red)("'loosely viewed'")` as change direction of count; given when you start counting you count increasingly further along the positive side of the number line!

This is why a double minus works. You change direction of count twice. That way you end up counting in the direction you started in!!