Question

How do you solve `|x + 4| > 10`?

Answer 1

See a solution process below:

Explanation:

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

`-10 > x + 4 > 10`

Now, subtract `color(red)(4)` from each segment of the system of inequalities to solve for `x` while keeping the system balanced:

`-10 - color(red)(4) > x + 4 - color(red)(4) > 10 - color(red)(4)`

`-14 > x + 0 > 6`

`-14 > x > 6`

Or

`x < -14`; `x > 6`

Or, in interval notation:

`(-oo, -14)`; `(6, oo)`

Answer 2

All values of `x` that are less than -14 `" "->" "x< -14`

All values of `x` that are more than 6 `" "->" "x>+6`

Explanation:

`color(blue)("Preamble")`
The two vertical lines represent something called an 'absolute value'
These are a special sort of brackets. The overall effect of which is that whatever the sign within them (positive or negative) the whole behaves as if it were positive

Example: `|-32| = |+32| = 32`

To emphasize a point with another example:

`" "-26color(white)(.) < color(white)(.) -14" "`

However `" "|-26| > |-14| " "` as it is the same as `+26>+14`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

Note that I represent some other positive value by `Delta`
`|x+4| > 10` is the same as: `|+-(10+Delta)|>10`

Lets determine the 'condition trigger values'.

Consider the trigger of `x+4=-10" "=>" "x= -14`
Consider the trigger of `x+4=+10" "=>" "x= +6`

So the domain (input) is all the value of:

All values of `x` that are less than -14 `" "->" "x< -14`

All values of `x` that are more than 6 `" "->" "x>+6`