SOLUTION: Find the solution set {{{(x-1)/(x+2)-(x-3)/(x+1)<1}}}

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Question 124678: Find the solution set

%28x-1%29%2F%28x%2B2%29-%28x-3%29%2F%28x%2B1%29%3C1

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-1%29%2F%28x%2B2%29-%28x-3%29%2F%28x%2B1%29%3C1 Start with the given inequality




Multiply both sides by the LCD %28x%2B2%29%28x%2B1%29%281%29. Doing this will eliminate every fraction.


%28x%2B1%29%28x-1%29-%28x%2B2%29%28x-3%29%3C%28x%2B2%29%28x%2B1%29 Distribute and multiply. Notice every denominator has been canceled out.


%28x%5E2-1%29-%28x%5E2-x-6%29%3Cx%5E2%2B3x%2B2 Foil


x%5E2-1-x%5E2%2Bx%2B6%3Cx%5E2%2B3x%2B2 Distribute the negative


x%5E2-1-x%5E2%2Bx%2B6-x%5E2-3x-2%3C0 Get every term to the left side


-x%5E2-2x%2B3%3C0 Combine like terms


If we set -x%5E2-2x%2B3 equal to zero and solve, we'll find that the critical values are x=-3 or x=1


Now let's test a value that is less than -3


Let x=-4

-x%5E2-2x%2B3%3C0 Start with the given inequality


-%28-4%29%5E2-2%28-4%29%2B3%3C0 Plug in x=-4


-5%3C0 Simplify


Since this inequality is true, this means that any values less than x=-3 will satisfy the inequality



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Now let's test a value that is in between -3 and 1


Let x=0

-x%5E2-2x%2B3%3C0 Start with the given inequality


-%280%29%5E2-2%280%29%2B3%3C0 Plug in x=0


3%3C0 Simplify


Since this inequality is not true, this means that any values in between -3 and 1 will not satisfy the inequality.


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Now let's test a value that is in greater than 1


Let x=2

-x%5E2-2x%2B3%3C0 Start with the given inequality


-%282%29%5E2-2%282%29%2B3%3C0 Plug in x=2


-5%3C0 Simplify


Since this inequality is true, this means that any values greater than x=1 will satisfy the inequality




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Answer:


So any values that are either less than -3 or greater than 1 will satisfy the inequality


In other words, x%3C-3 or x%3E1


So the solution in interval notation is

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