<s>[INST] Irrational inequalities question: $\sqrt { -3x+1 } + \sqrt {6x+1} < \sqrt {3x+4}$ and $\sqrt { -6x+10 } + \sqrt {-x+2} \gt \sqrt {4x+5}$ Consider the following inequalities: 
\begin{equation*}
 \sqrt { -3x+1 } + \sqrt {6x+1} \lt \sqrt  {3x+4}, \\
 \sqrt { -6x+10 } + \sqrt {-x+2} \gt \sqrt {4x+5}.
\end{equation*}
Attempt at a solution; after performing all the standard actions, I got: $\frac 16\lt x \le\frac 13 $  or  $\frac {-1}{6}\le x \lt 0 $, for first inequality, and:  $\frac {-34}{97}\lt x \lt 1 $  for the second. 
Official results state however:  $\frac {-1}{6}\le x \lt 0 $  for first inequality, and
 $\frac {-5}{4}\le x \lt 1 $ for the second. 
What was my mistake? 
 [/INST] The second equation became $ 97x^2-66x-31<0$
$(97x+31)(x-1)<0$
Therefore
$$-\frac{31}{97}<x<1$$
This condition satisfies all conditions.
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