So the question is,
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:
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!
For what values of x is 1/(1 + x) > -1?
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:
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!