Values for p so inequality is right for every x. − 9 &lt; 3 x 2 + p x − 6 x 2 − x + 1 &lt; 6I have done problems where I need to get the parameter so the inequality is right for real values of x. But what do I have to do to make it right for every value of x? What condition? I am not asking for anyone to throw me the answer of the whole thing. Just to clarify what the condition needs to be so I can finish on my own.

Question

Values for p so inequality is right for every x.   −  9  &amp;lt;            3              x        2            +      p      x      −      6                      x        2            −      x      +      1        &amp;lt;  6I have done problems where I need to get the parameter so the inequality is right for real values of x. But what do I have to do to make it right for every value of x? What condition? I am not asking for anyone to throw me the answer of the whole thing. Just to clarify what the condition needs to be so I can finish on my own.

Haylee Krause · Accepted Answer

Since       x    2    −  x  +  1  &amp;gt;  0, we obtain  3      x    2    −  (  p  +  6  )  x  +  12  &amp;gt;  0and  12      x    2    +  (  p  −  9  )  x  +  3  &amp;gt;  0  ,for which we need  (  p  +  6      )    2    &amp;lt;  144and  (  p  −  9      )    2    &amp;lt;  144  ,which gives  −  18  &amp;lt;  p  &amp;lt;  6and  −  3  &amp;lt;  p  &amp;lt;  21  ,which is  −  3  &amp;lt;  p  &amp;lt;  6.

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