** I kind of went a little over the top and solved the whole problem for you, if you don't want the whole answer stop reading when you are satisfied**

The problem is that you're divining the number line incorrectly. I'm not sure where you learnt to substitute for the zeros but this is the way I learnt so I'll be happy to explain more if needed.

There are three possible cases:

• abs(x+5), abs(x+3) + 1 are either both positive or negative.
• abs(x+5) is negative, abs(x+3) + 1 is positive.
• abs(x+5) is positive, abs(x+3) + 1 is negative.

Case 1:

Note we can ignore the abs() as we have assumed they're both of same sign-age
(x + 5) / [(x + 3) + 1] > 1 
(x + 5) > (x + 3) + 1
x + 5 > x + 3 + 1
       0 > 1

This answer doesn't make sense so this can't be the right solution.

Case 2: abs(x + 5) is negative.

-(x + 5) / [(x + 3) + 1] > 1
-x - 5 > x + 3 + 1
-x - 5 > x + 4
     -1 > 2x
   -1/2 > x

So x < -1/2 is a possible point where the inequality changes.

Case 3: abs(x + 3) is negative.

x + 5 / [ -(x+3) + 1 ] > 1
x + 5 > -(x + 3) + 1
x + 5 < -x - 3 + 1
2x < -7
x < -7/2

So another point which could be interesting is x < -7/2

Now we have two x coordinates (x = -7/2, x = -1/2) where the inequality may change from True to False (or visa versa). Or in as ranges of x we want to consider are (-inf, -7/2), [-7/2, -1/2), [-1/2, inf).

Note that I haven't written them in bracket notation as I have calculated the values as equalities because it makes it less likely for me to make a mistake on how the equality sign is facing.

So now we test a point within each region:

(-inf, -7/2):

Try x = -10 (This can be any number within that region)
abs(-10 + 5) / [abs(-10 + 3) + 1] > 1
5 / [7 + 1] > 1
5 / 8 > 1
False

[-7/2, -1/2):

Try x = -1
abs(-1 + 5) / [abs(-1 + 3) + 1] > 1
4 / 3 > 1
True

[-1/2, inf):

Try x = 0 (Try take easy numbers so trying x = 21325121412 isn't a smart choice)
abs(5) / [abs(3) + 1] < 1
5 / 4 > 1
True

Therefore the equation is False for x < -7/2 and True for x >= -7/2/