Now, one way to do this is to write the inequality relative to zero, and then find the behavior of the graph relative to the zeroes of the numerator and denominator:For what values of x is 1/(1 + x) > -1?

1/(1 + x) > -1 => 1/(1+x) + 1> 0 => 1/(1+x) + (1+x)/(1+x) > 0

=> (2+x)(1+x) > 0, and we simply find what happens when x < -2, -2 < x < -1, and x > -1.

What I don't understand is why I can't answer this by multiplying through by 1 +x if I consider both a) 1 +x >0 and b) 1+x < 0.

Here's the attempt:

Case a): 1 + x > 0 => x > -1, so

1/(1+x) > -1

=> (1+x) (1)/(1+x) > -1 (1+x) (we can multiply both sides by 1+x because we assume 1+x is positive)

=> x > -2.

Hence, the inequality is true when x > -2 and x > -1--so x > -1.

Case b): 1+x < 0 => x < -1, so

1/(1+x) > -1

=> (1+x) 1/(1+x) < -1 (1+x) (we assume 1+x is negative, so we reverse the sign)

=> 1 < 1 + x

=> x > 0.

Hence, the inequality is true when x < -1 and x > 0--this is never true.

So the real answer is x > -1 or x <-2, and I think the reason my attempt at multiplying through by an unknown doesn't work is that, in my assumption x in (-1, infinity) clearly includes both negative and positive values, so we can't say what the value of x will be when 1 + x > 0, and that leads the answer to be wrong.

But I'm not 100 percent certain it's impossible, and would like some help!