   SEARCH HOME Math Central Quandaries & Queries  Question from Nazrul, a teacher: What is the solution of absolute vaue of (1/(x+1))<1? Please explain. I use the vertical bars like parentheses for absolute value. For example, |x| means "absolute value of x" and 5x + |x - 2| means "5x plus the absolute value of (x - 2)".

I don't want to do all your work for you, so I'll change the numbers and show you how it is done:

|8 / (1 - x) | < 4

The numerator 8 doesn't affect the sign (multiplying a positive factor leaves a negative as negative or a positive as positive). So I can move the absolute value symbols to just the denominator.

8 / |1 - x | < 4

I can multiply both sides by |1 - x| which is never negative (it is zero or positive), therefore the direction of the inequality doesn't change:

8 < 4 | 1 - x |

Divide by 4:

2 < | 1 - x |

Now I have to split it up into positive and negative.

2 < 1 - x or 2 < - (1 - x)

and solve them separately:
2 < 1 - x reduces to 1 < -x, so we multiply by -1 and change the direction: -1 > x.
2 < - (1 - x) reduces to 3 < x.

So either x < -1 or x > 3.

Hope this helps,
Stephen La Rocque     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.