How to solve inequality problem without factoring or quadratic equation

Question

I'm tutoring someone, and I'm stuck on one of her problems. The equation is $\sqrt{x+14}\le x-16$. She hasn't been taught the quadratic formula or how to factor these problems yet. Is there a way to solve this problem without using either of these methods? It's possible that the teacher wrote the problem without checking to make sure that it is solvable with what they know, but I want to make sure.

Answer 1

it may be easier to make a change of variable $u = x + 14, x = u - 14.$ now we have $$\sqrt u \le u - 30$$ the equality is satisfied by $$u = 36$$ and the inequality for $$ u > 36$$ this can be argued by picking test points on either side of $u = 36.$ what we have now is $$\sqrt u \le u - 30 \text{ for } u \ge 36.$$ we can now translate it back to the original $x$ variable $$ \sqrt{x+14} \le x - 16 \text{ for } x \ge 22.$$