Question

How do you solve the equation `6+sqrt(2x+11)=-x`?

Answer 1

`x in O/`

Explanation:

For starters, you know that when working with real numbers, you can only take the square root of a positive number, so you can say that you must have

`2x + 11 >= 0 implies x >= -11/2`

Moreover, the square root of a positive number can only return a positive number, so you also know that

`sqrt(2x + 11) >= 0`

This implies that

`overbrace(6 + sqrt(2x + 11))^(color(blue)(>= 6)) = -x`

So you can say that

`-x >= 6 implies x <= -6 or x in (-oo, -6]`

However, you already know that you need to have `x >= -11/2`, or `x in [-11/2, + oo)`.

Since

`(-oo, - 6] nn [-11/2, + oo) = O/`

you can say that the original equation has no real solution, or `x in O/`.