Question

How do you graph and solve `|2x-6|>|x+4|`?

Answer 1

`x < 2/3color(white)("XX")orcolor(white)("XX")x > 10`

(graphed as a number line)

Explanation:

If `abs(2x-6) > abs(x+4)`
then
`color(white)("XXX")(2x-6)^2 > (x+4)^2`

`color(white)("XXX")4x^2-24x+36 > x^2+8x+16`

`color(white)("XXX")3x^2-32x+20 > 0`

`color(white)("XXX")(3x-2)(x-10) > 0`

This can only be true if:
[1] both factors are greater than `0`
`color(white)("XXX")"if " x > 2/3 " "underline("and")" " x > 10rarr x > 10`
or
[2] both factors are less than `0`
`color(white)("XXX")"if " x < 2/3 " "underline("and")" " x < 10 rarr x <2/3`