Find all the real values of k for which the following system of inequalities − 3 &lt; x 2 + k x − 2 x 2 − x + 1 &lt; 2is fullfilled for all real values of xI have attempted this question but my teacher told me that we have to take Δ &lt; 0 in − 3 ( x 2 − x + 1 ) &lt; x 2 + k x − 2The answer is to be provided in intervals

Question

Find all the real values of   k for which the following system of inequalities  −  3  &amp;lt;                    x        2            +      k      x      −      2                      x        2            −      x      +      1        &amp;lt;  2is fullfilled for all real values of   xI have attempted this question but my teacher told me that we have to take   Δ  &amp;lt;  0 in   −  3  (      x    2    −  x  +  1  )  &amp;lt;      x    2    +  k  x  −  2The answer is to be provided in intervals

Leland Ochoa · Accepted Answer

Continue where you left off:   4      x    2    +  (  k  −  3  )  x  +  1  &amp;gt;  0  ,  ∀  x  ∈  R    ⟹        △    1    &amp;lt;  0    ⟹    (  k  −  3      )    2    −  16  &amp;lt;  0    ⟹    (  k  −  7  )  (  k  +  1  )  &amp;lt;  0    ⟹    −  1  &amp;lt;  k  &amp;lt;  7and similarly the other one gives:      △    2    =  (  k  +  2      )    2    −  16  &amp;lt;  0    ⟹    (  k  +  6  )  (  k  −  2  )  &amp;lt;  0    ⟹    −  6  &amp;lt;  k  &amp;lt;  2. Combine the two inequalities, you have:  −  1  &amp;lt;  k  &amp;lt;  2

manierato5h · Accepted Answer

I guess   Δ  &amp;lt;  0 as the whole quadratic   4      x    2    +  (  k  −  3  )  x  +  1  &amp;gt;  0 so the parabola won&#039;t be touching the x - axis.

Find all the real values of k for which the following system of inequalities −<!-- −

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