Question

How do you solve `abs(2x-4)>10`?

Answer 1

`color(green)(x < -3)color(white)("XX")orcolor(white)("XX")color(green)(x > 7)`

Explanation:

Here are two methods you might use to solve this:

Method 1: Start by squaring both sides
`abs(2x-4) > 10`

`color(white)("XXX")rarr (2x-4)^2 > 10^2`

`color(white)("XXX")4x^2-16x+16 > 100`

`color(white)("XXX")x^2-4x-21 > 0`

`color(white)("XXX")(x-7)(x+3) > 0`

This will be true it both `(x-7)` and `(x+3)` have the same sign.
That is if
`color(white)("XXX")(x-7) < 0 rarr x < 7`
`color(white)("XXX")`and
`color(white)("XXX")(x+3) < 0 rarr x < -3`
`color(white)("XXXXXXXXX")`for both to be true: `x < -3`
or
`color(white)("XXX")(x-7) > 0 rarr x > 7`
`color(white)("XXX")`and
`color(white)("XXX")(x+3) > 0 rarr x > -3`
`color(white)("XXXXXXXXX")`for both to be true: `x > 7`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Method 2: Separate the negative and positive versions of `(2x-4)` immediately
`abs(2x-4) > 0`

#{: ("if " (2x -4) < 0,color(white)("X")orcolor(white)("X"),"if " (2x-4) > 0), (abs(2x-4) > 10 rarr 4-2x > 10,,abs(2x-4) > 10 rarr 2x-4 > 10), (color(white)("XXXXXXXX")rarr -2x > 6,,color(white)("XXXXXXXX")rarr x > 7), (color(white)("XXXXXXXX")rarr x < -3,,) :}#

Answer 2

`x<-3` or `x>7`

Explanation:

`|2x-4|>10`

If `|2x-4|` is negative,

`-(2x-4)>10`
`color(white)(,)-2x+4>10`
`color(white)(xxx.)-2x>6`
`color(white)(xxxxx.)2x<-6`
`color(white)(xxxxx..)x<-3`

If `|2x-4|` is positive,

`2x-4>10`
`color(white)(xxx)2x>14`
`color(white)(xxx.)x>7`

Hence `x<-3` or `x>7`.