SOLUTION: Solve the following inequality 3 2 2 ------ < ------ < ------- 4x+3 3 x+5

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Question 1160786: Solve the following inequality
3 2 2
------ < ------ < -------
4x+3 3 x+5
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
3%2F%284x%2B3%29+%3C++2%2F3++%3C+2%2F%28x%2B5%29

if a+%3C+u%3C+b then a%3Cu and+u+%3C+b

3%2F%284x%2B3%29+%3C++2%2F3 and 2%2F3++%3C+2%2F%28x%2B5%29

first solve

3%2F%284x%2B3%29+%3C++2%2F3.........subtract 2%2F3 from both sides
3%2F%284x%2B3%29+-2%2F3%3C++2%2F3-2%2F3
3%2F%284x%2B3%29+-2%2F3%3C+0......common denominator is 3%284x%2B3%29
9%2F3%284x%2B3%29+-2%284x%2B3%29%2F3%284x%2B3%29%3C+0
%289+-2%284x%2B3%29%29%2F3%284x%2B3%29%3C+0
%289+-8x-6%29%2F3%284x%2B3%29%3C+0
%283+-8x%29%2F3%284x%2B3%29%3C+0 ..........find solutions of both numerator and denominator
%283+-8x%29%3C+0
highlight%28x%3E3%2F8%29
or
3%284x%2B3%29%3C+0
%284x%2B3%29%3C+0
4x%3C+-3
highlight%28x%3C+-3%2F4%29


and, now solve 2%2F3++%3C+2%2F%28x%2B5%29
0++%3C+2%2F%28x%2B5%29-2%2F3
0++%3C+6%2F3%28x%2B5%29-2%28x%2B5%29%2F3%28x%2B5%29
0++%3C+%286-2%28x%2B5%29%29%2F3%28x%2B5%29
0++%3C+%286-2x-10%29%2F3%28x%2B5%29
0++%3C+%28-2x-4%29%2F3%28x%2B5%29
solutions:
0++%3C+-2x-4
2x++%3C-4
highlight%28x++%3C-2%29
or
0++%3C3%28x%2B5%29
0++%3Cx%2B5
-5+%3Cx
highlight%28x%3E-5%29


so, the intervals that satisfy the required condition < 0 are:
highlight%28x%3E3%2F8%29 or highlight%28x%3C+-3%2F4%29
and
highlight%28-5%3Cx%3C-2%29
Merge overlapping intervals which is your solution:
highlight%28red%28-5%3Cx%3C-2%29%29