Question

How do you solve `2| 2x - 7| + 6> 8`?

Answer 1

`x<3` or `x>4`

Explanation:

First isolate the absolute value portion:

Subtract `6` from both sides:

`2abs(2x-7)>2`

Divide both sides by `2`:

`abs(2x-7)>1`

Since we're dealing with an absolute value, recall that `abs(-5)=5`. So, if `2x-7<-1`, then `abs(2x-7)>1`.

We can split up the absolute value into a negative and positive version.

The negative version we already stated was:

`2x-7<-1" "=>" "x<3`

And the normal version. if `2x-7` is positive, is:

`2x-7>1" "=>" "x>4`

Answer 2

The two value sets for `x` are
`x<3`
`x>4`

Explanation:

Given:

`" "color(green)(2|2x-7|+6" ">" "8)`

Multiplye both sides by `color(red)(1/2)`

`" "color(green)(2/(color(red)(2))|2x-7|+6/(color(red)(2))" ">" " 8/(color(red)(2)))`

`" "|2x-7|+3" ">" "4`

Subtract 3 from both sides

`" "|2x-7|" " >" "1`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Just for the moment, consider the example `|2x-7|=1` then we would have `|+-1|=1` so in this case `2x-7=-1 and 2x-7=+1`

However, we have `|2x-7|>1` so the feasible values are such that `2x-7` has to be more negative than -1 and `2x-7` has to be more positive than +1

Consider the case `2x-7<-1 => 2x<6 => x<3`

Consider the case `2x-7>+1=>2x>8=>x>4`

`color(blue)("So the solutions for this inequality are: "x<3 and x>4)`