Question

Solve 1-x/x+5>2-x/x+6 without the use of a graphing calculator show all your work.

Answer 1

the way you typed it, the statement would simply become:
1-x/x+5>2-x/x+6
1 - 1 + 5 > 2 - 1 + 6
5 >7 , which is of course false

so you must have meant:
(1-x)/(x+5) > (2-x)/(x+6) , x ≠ -5,-6
(1-x)(x+6) > (x+5)(2-x)
-x^2 - 5x + 6 > -x^2 - 3x + 10
-2x > 4
x < -2 , x ≠ -5,-6

or
(1-x)/(x+5) > (2-x)/(x+6), x ≠ -5,-6
(1-x)/(x+5) - (2-x)/(x+6) > 0
[(1-x)(x+6) - (2-x)(x+5)]/[(x+5)(x+6)] > 0
-x^2 - 5x + 6 - (-x^2 - 3x + 10) > 0
-2x - 4 > 0
2x + 4 < 0
x < -2 , x ≠ -5,-6

Answer 2

when we get to
-x^2 - 5x + 6 - (-x^2 - 3x + 10) > 0
this is true only if (x+5)(x+6) > 0
That is, x < -6 or x > -5
so, combining that with x < -2, we have a solution set of
(-∞,-6)U(-5,-2)