Question

How do you solve and write the following in interval notation: `3x+4≥5x-8`?

Answer 1

`x in (-oo,6]`

Explanation:

Remember you can add/subtract the same amount and multiply/divide by any amount greater than zero to both sides of an inequality without invalidating the inequality.

`3x+4 >= 5x-8`
`color(white)("XXX")3x+12 >= 5x`

`color(white)("XXX")12 >= 2x`

`color(white)("XXX")6 >= x`

or (reversing so `x` is on the left, which is more common)
`color(white)("XXX")x <= 6`

Answer 2

You may add or subtract on both sides of the inequality.

Explanation:

Add `8` to both sides:
`3x+4+8>=5x-cancel8+cancel8->3x+12>=5x`#

Subtract `3x` from both sides:
`cancel(3x)-cancel(3x)+12>=5x-3x->12>=2x->`

Divide by `2`, a positive number:
`6>=x->x<=6`

`->x in [6,oo)`, where `[` means 'inclusive'.

Note: you may be used to a different notation, but I think the above is clear enough to translate.