Question 904505
I've marked all of the incorrect answers in red boxes. I've also labeled the red boxes (A through G) to reference them below


<img src = "http://i150.photobucket.com/albums/s91/jim_thompson5910/error2a_zps20cf7c08.png">

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Explanation for Box A (column 1): 

Pick a number that is in the interval (-infinty, -4). I'll pick x = -5

x+4 = -5+4 = -1 which is negative, therefore x+4 < 0 on the interval (-infinty, -4)

Further proof: x+4 < 0 ------> x < -4 

So you should have a negative sign in box A.

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Explanation for Box B (column 1): 

The signs above this last box in column 1 should be: +, -, - from top to bottom in that order

The two negatives multiply and/or divide to get a positive, so we're left with positive * positive = positive

You should have a "+" in box B

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Explanation for Box C (column 4): 

plug x = -1 into x+1 to get x+1 = -1+1 = 0

The number 0 should be in box C

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Explanation for Box D (column 4): 

since x+1 is zero when x = -1, and x+1 is in the numerator, the whole expression is 0

you should have a 0 in box D
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Explanation for Box E (column 5): 

x = 0 is in the interval (-1,7)

7 - x = 7 - 0 = 7

So you have have a plus sign in box E

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Explanation for Box F (column 5): 

x = 0 is in the interval (-1,7)

x+1 = 0+1 = 1

You should have a plus sign in box F

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Explanation for Box G (column 5): 

The expression/function is defined on the interval (-1,7). It is only undefined when x = -4 since this causes a division by zero error.

In the boxes above box G, all of them are filled with a + sign

So the overall function is positive on the interval (-1,7)

You should have a + in box G

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Here is what the chart should look like. My corrections are in blue


<img src = "http://i150.photobucket.com/albums/s91/jim_thompson5910/error3_zpsd30453ba.png">


And here is the graph of *[Tex \Large  f(x) = \frac{(7-x)(x+1)}{x+4}] (the dashed blue line is the vertical asymptote)


<img src = "http://i150.photobucket.com/albums/s91/jim_thompson5910/9-22-20143-02-08PM_zps3e6c6261.png">


This confirms the sign chart. Using either the graph or the sign chart, we see that the answer to *[Tex \Large  \frac{(7-x)(x+1)}{x+4} \le 0] in interval notation is *[Tex \Large  (-4,-1) \cup (7,\infty)]