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# Help plzz

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$${x^2+x+3 \over 2x^2+x-6}{ \ge0}$$            Find all values of x such that

Mar 2, 2020

#1
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Note  that    the  discriminant of  x^2  + x  + 3 =    1^2  -  4(1) (3)  =  1  - 12 =   -11

This means that   this   function   has no real roots.....and since it's a parabola that turns upward.....then it  never  intersects the  x axis, so it is   never   <  0   for  any  value  of  x

So.....we  only  need to  find  the  values   that   make   2x^2  + x   - 6   <  0

So

2x^2  + x  - 6    <  0   factor   the  left side

(2x  - 3) ( x + 2)     <   0

Note  that      on   (inf, -2)   this  function  is  >  0

And on  ( 3/2 , inf)   the same is true

So....it's  only  <  0    on   (  -2,  3/2)

So.....going back to the original  function...it  must  be  ≥  0    on      (-inf , -2)   U ( 3/2 , inf)

See the graph here :  https://www.desmos.com/calculator/agjohh32v6   Mar 2, 2020
#2
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Thanks CPhill, you are the best!

Mar 2, 2020
#3
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Not the  best.....I just know a few things.....LOL!!!   CPhill  Mar 2, 2020