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So task is: $\sqrt{-x^2+x+6} > 1-x$... I know the principle of solving tasks with this, but at this task here I have $\;-x^2+x+6$ under sqrt and this makes problem for me in forming system and solving it... I did this: https://ibb.co/prrDMrr

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closed as off-topic by Peter, José Carlos Santos, Song, Cesareo, Lee David Chung Lin Mar 15 at 0:43

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  • 4
    $\begingroup$ Please typeset your equations using MathJax. This is unreadable. $\endgroup$ – Yves Daoust Mar 14 at 21:28
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    $\begingroup$ Welcome to Math.SE! I am having trouble reading the equations that you included in your question. Please format your questions using MathJax. This page should give you a start at learning how to typeset mathematics here so that your posts say what you want them to, and also look good. $\endgroup$ – Brian Mar 14 at 21:28
  • $\begingroup$ Thanks, I am literally on my phone right now and I don't have access to my computer, so thanks Bernard for editing :) I will read that page and use these functions if I have any questions in future. $\endgroup$ – Pshyotic Mar 14 at 21:35
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Hint:

The domain of validity of the inequation is defined by the condition $$ x^2-x-6\le 0$$ This quadratic polynomial has two integer roots: $3$ and $-2$, hence the domain of validity is the interval $[-2,3]$.

Now, on its domain of validity, $$\sqrt A >B\iff (B<0)\:\text{ or } (A>B^2).$$

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  • $\begingroup$ don't we have two main types of irrational inequalities? One with: sqrt(f(x)) < g(x) and sqrt(f(x)) > g(x) and based on that we decide either we need to form one or two systems for solving this? I know how to fond x1 and x2 by Viet's formulas... $\endgroup$ – Pshyotic Mar 14 at 21:45
  • $\begingroup$ This one is of the second type, but anyway you have to determine the domain of validity. $\endgroup$ – Bernard Mar 14 at 21:50
  • $\begingroup$ So now if I understand I should make three statments for the first system, where I will determine where they are interconnected(in number axis) , and to do the same for the other system, and find a solution which will be union of all of these solutions? $\endgroup$ – Pshyotic Mar 14 at 21:58

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