<s>[INST] Inequality with square root $x+\sqrt{x^2-10x+9}\ge \sqrt{x+\sqrt{x^2-10x+9}}$ Good morning to everyone! The inequality is the following:$$ x+\sqrt{x^2-10x+9}\ge \sqrt{x+\sqrt{x^2-10x+9}} $$. I don't know how to solve it. Here's what I tried: $$x+\sqrt{x^2-10x+9}\ge \sqrt{x+\sqrt{x^2-10x+9}} \rightarrow 2x^2-11x+9+\sqrt{x^2-10x+9}\left(2x+1\right)\ge \:\sqrt{x^2-10x+9}$$ Therefore $ 2x+1 \ge 0 => x\ge \frac{1}{2} $ so $x$ belongs to $(\frac{1}{2},\infty) $ and $ x^2-10x+9 \ge 0 => x$ belongs to $ (-\infty,1] $ and $[9,\infty) $. Therefore the statements are contradicting each other.
 [/INST] The suggestion about $u\ge\sqrt{u}$ helps in removing one level of square roots. Now you have
$$
x+\sqrt{x^2-10x+9}\ge1
$$
or
$$
\sqrt{x^2-10x+9}\ge1-x
$$
If you notice that the expression under the square root is $(x-1)(x-9)$, you can possibly use the clever trick shown in another answer. On the other hand, such an inequality can be analyzed in a systematic way, by transforming it into two systems of inequalities:
$$
\mathrm{(I)}
\begin{cases}
x^2-10x+9\ge0 \\[4px]
1-x<0
\end{cases}
\qquad\text{or}\qquad
\mathrm{(II)}
\begin{cases}
\bigl[\,x^2-10x+9\ge0\,\bigr] \\[4px]
x^2-10x+9\ge(1-x)^2 \\[4px]
1-x\ge0
\end{cases}
$$
The first system comes from the consideration that the square root is non-negative as soon as it exists, so it will be greater than a negative number; the second system comes from the consideration that an inequality between non-negative numbers is equivalent to the same inequality between their squares.
System (I) can be transformed into
$$
\mathrm{(I)}
\begin{cases}
x\le1 \quad\text{or}\quad x\ge9 \\[4px]
x>1
\end{cases}
$$
which has $x\ge9$ as solution set.
System (II), where the top inequality is implied by the middle one, becomes
$$
\mathrm{(II)}
\begin{cases}
x^2-10x+9\ge1-2x+x^2 \\[4px]
x\le1
\end{cases}
$$
and it is easily seen that the solution set is $x\le1$.
Summing up, the solution set of the original inequality is
$$
\boxed{\quad\vphantom{\Big|}x\le 1\quad\text{or}\quad x\ge9\quad}
$$
</s>