SOLUTION: Rearrange the first inequality to match the second inequality and fill in the blanks for the variables. Then select the appropriate intervals to create the solution set for this in

Algebra ->  Inequalities -> SOLUTION: Rearrange the first inequality to match the second inequality and fill in the blanks for the variables. Then select the appropriate intervals to create the solution set for this in      Log On


   

Question 911693: Rearrange the first inequality to match the second inequality and fill in the blanks for the variables. Then select the appropriate intervals to create the solution set for this inequality.
10%2Fx+%2B+3%2F%28x%2B1%29+%3C+13%2F%28x%2B2%29
%28____x+%2B+____+%29%2F+%28x%28x%2B1%29%28x%2B2%29%29+%3C+0
Then create appropriate intervals.
Thank you
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Check my arithmetic (I have been known to make mistakes).
{{10/x + 3/(x+1) < 13/(x+2)}}} ---> ---> ---> --->

--> ---> %2813x%5E2%2B36x%2B20-%2813x%5E2%2B13x%29%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3C0 --->

%2813x%5E2%2B36x%2B20-13x%5E2-13x%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3C0 --> %2823x%2B20%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3C0
The factors of numerator and denominator are
23x%2B20 , x , x%2B1 and x%2B2 .
Those factors become zero at
x=-20%2F23 , x=0 , x=-1 and x=-2 respectively,
with -2%3C-1%3C-20%2F23%3C0
Each factors becomes zero and changes sign at one of those values of x.
For greater values of x, the factor is positive.
For lesser values of x, the factor is negative.

So, for x%3C-2 all four factors are negative,
and %2823x%2B20%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3E0 .
For highlight%28-2%3Cx%3C-1%29 only one of the four factors is positive,
and %2823x%2B20%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3C0 .
For -1%3Cx%3C-20%2F23 two of the four factors are positive (the other two are negative),
and %2823x%2B20%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3E0 .
For highlight%28-20%2F23%3Cx%3C0%29 three of the factors are positive (only x is negative),
and %2823x%2B20%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3C0 .
Finally, for x%3E0 , all four factors are positive,
and %2823x%2B20%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3C0 .

So the solution to
%2823x%2B20%29%2F%28x%28x%2B1%29%28x%2B2%29%29%3C0 and to {{10/x + 3/(x+1) < 13/(x+2)}}} is
highlight%28-2%3Cx%3C-1%29 or highlight%28-20%2F23%3Cx%3C0%29 .