Question

How do you solve and write the following in interval notation: `-2(x-1)-12<2(x+1)`?

Answer 1

See a solution process below:

Explanation:

First, expand the terms on both sides of the inequality by multiplying each term within the parenthesis by the term outside the parenthesis:

`color(red)(-2)(x - 1) - 12 < color(blue)(2)(x + 1)`

`(color(red)(-2) xx x) - (color(red)(-2) xx 1) - 12 < (color(blue)(2) xx x) + (color(blue)(2) xx 1)`

`-2x - (-2) - 12 < 2x + 2`

`-2x + 2 - 12 < 2x + 2`

`-2x - 10 < 2x + 2`

Next, add `color(red)(2x)` and subtract `color(blue)(2)` from each side of the inequality to isolate the `x` term while keeping the inequality balanced:

`color(red)(2x) - 2x - 10 - color(blue)(2) < color(red)(2x) + 2x + 2 - color(blue)(2)`

`0 - 12 < (color(red)(x) + 2)x + 0`

`-12 < 4x`

Now, divide each side of the inequality by `color(red)(4)` to solve for `x` while keeping the inequality balanced:

`-12/color(red)(4) < (4x)/color(red)(4)`

`-3 < (color(red)(cancel(color(black)(4)))x)/cancel(color(red)(4))`

`-3 < x`