SOLUTION: 6x^2-5x-1/12x^2-11x+2 is less than 0

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Question 362462: 6x^2-5x-1/12x^2-11x+2 is less than 0
Found 2 solutions by stanbon, Fombitz:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
6x^2-5x-1/12x^2-11x+2 is less than 0
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Factor:
[(6x+1)(x-1)]/[(3x-2)(4x-1)] < 0
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Find the boundary values for the inequality
which are the solutions to the EQUALITY.
Those values are x = -1/6 and x = 1
Also take note of indeterminate values.
Those are x = 2/3 and x = 1/4
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Draw a number line and plot the boundary values:
....-1/6....1/4....2/3....1....
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To find the solutions for the inequality, test
a value from each of the three bounded intervals
in the inequality.
[(6x+1)(x-1)]/[(3x-2)(4x-1)] < 0
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If x = -1 you get: (-*-)/(-*-) < 0; this is false
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If x = 0 you get: (+*-)/(-*-) < 0: true, so solutions in (-1/6,1/4)
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Do the same for x = 1/3 , for x = 5/6, and x = 1
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Cheers,
Stan H.

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
%286x%5E2-5x-1%29%2F%2812x%5E2-11x%2B2%29%3C0
%28%28x-1%29%286x%2B1%29%29%2F%28%283x-2%29%284x-1%29%29%3C0
1, -1/6, 2/3, 1/4
Break up the number line into 5 regions using the critical points of the function.
Region 1:(-infinity,-1%2F6)
Region 2:(-1%2F6,1%2F4)
Region 3:(1%2F4,2%2F3)
Region 4:(2%2F3,1)
Region 5:(1,infinity)
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For each region, choose a point in the region (not an endpoint).
Test the inequality.
If the inequality is satisfied, the region is part of the solution region.
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Region 1:x=-1
%28%28x-1%29%286x%2B1%29%29%2F%28%283x-2%29%284x-1%29%29%3C0
%28%28-1-1%29%28-6%2B1%29%29%2F%28%28-3-2%29%28-4-1%29%29%3C0
%28%28-2%29%28-5%29%29%2F%28%28-5%29%28-5%29%29%3C0+
2%2F5%3E0+
False, Region 1 is not part of the solution region.
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Region 2:x=0
%28%28x-1%29%286x%2B1%29%29%2F%28%283x-2%29%284x-1%29%29%3C0
%28%28-1%29%281%29%29%2F%28%28-2%29%28-1%29%29%3C0
-%281%2F2%29%3C+0+
True, Region 2 is part of the solution region.
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Region 3:x=1%2F2
%28%28x-1%29%286x%2B1%29%29%2F%28%283x-2%29%284x-1%29%29%3C0
%28%281%2F2-1%29%283%2B1%29%29%2F%28%283%2F2-2%29%282-1%29%29%3C0
%28%28-1%2F2%29%284%29%29%2F%28%28-1%2F2%29%281%29%29%3C0+
4%3C0+
False, Region 3 is not part of the solution region.
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Region 4:x=3%2F4
%28%28x-1%29%286x%2B1%29%29%2F%28%283x-2%29%284x-1%29%29%3C0
%28%283%2F4-1%29%289%2F2%2B1%29%29%2F%28%289%2F4-2%29%283-1%29%29%3C0
%28%28-1%2F4%29%2811%2F2%29%29%2F%28%281%2F4%29%282%29%29%3C0+
-11%2F4%3C0+
True, Region 4 is part of the solution region.
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Region 5:x=2
%28%28x-1%29%286x%2B1%29%29%2F%28%283x-2%29%284x-1%29%29%3C0
%28%282-1%29%2812%2B1%29%29%2F%28%286-2%29%288-1%29%29%3C0
%28%281%29%2813%29%29%2F%28%284%29%287%29%29%3C0+
13%2F28%3C0+
False, Region 5 is not part of the solution region.
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Solution Region: (-1%2F6,1%2F4) U (2%2F3,1)
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Graphical verification: Look for regions where the function is below the x-axis (y%3C0)
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