SOLUTION: help Solve the inequality. x-5 < 1 ___ x+4

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Question 616627: help Solve the inequality.
x-5 < 1
___
x+4
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
help Solve the inequality.
%28x-5%29%2F%28x%2B4%29 < 1 

Subtract 1 from both sides:

%28x-5%29%2F%28x%2B4%29 - 1 < 0

%28x-5%29%2F%28x%2B4%29 - 1%2F1 < 0

Get an LCD of (x+4) and multiply the 1%2F1 by %28x%2B4%29%2F%28x%2B4%29 < 0

%28x-5%29%2F%28x%2B4%29 - 1%2F1·%28x%2B4%29%2F%28x%2B4%29 < 0

%28x-5%29%2F%28x%2B4%29 - %28x%2B4%29%2F%28x%2B4%29 < 0

Combine the numerators over the LCD:

%28%28x-5%29-%28x%2B4%29%29%2F%28x%2B4%29 < 0

%28x-5-x-4%29%2F%28x%2B4%29 < 0

%28-9%29%2F%28x%2B4%29 < 0

To find the critical number, set x+4 = 0, getting x = -4

Since there is only one critical number we have only two
intervals to test:

Interval          | (-oo,-4)  |  (-4, oo) |
Test value        |    -5     |      0    |
Sign of left side |           |           |

The left side is  %28-9%29%2F%28x%2B4%29

We substitute test value -5 in %28-9%29%2F%28x%2B4%29
%28-9%29%2F%28-5%2B4%29 = 9 which is positive so we put a +

Interval          | (-∞,-4)  |  (-4, ∞) |
Test value        |    -5    |      0   |
Sign of left side |     +    |          |

We substitute test value 0 in %28-9%29%2F%28x%2B4%29
%28-9%29%2F%280%2B4%29 = -9%2F4 which is negative so we put a -

Interval          | (-∞,-4)  |  (-4, ∞) |
Test value        |    -5    |      0   |
Sign of left side |     +    |      -   |

Since it's < 0 we choose the interval (-4, ∞) as the solution

Edwin