Thread: inequality involving a fraction

x-1/x+2> or = 0

Firslty you want to identify the points at which the expression is equal to zero. That is, solve x-1/x+2 = 0. Multiplying up by x gives x^2 + 2x - 1 = 0, which has solutions x = -1 + sqrt(2) and -1 - sqrt(2), approximately 0.414 and -2.414 respectively. This particular expression also has a vertical asymptote, a polite way of saying there's an infinity, at x=0. So the expression isn't going to cross the x-axis, that is, isn't going to change sign, except at these three points. So let's see what its values are like in the regions

x < -1-sqrt(2) for example x = -10 value = -7.9 sign is -
x = -1-sqrt(2) value = 0
-1-sqrt(2) < x < 0 for example x = -1 value = 4 sign is +
x = 0 value undefined
0 < x < -1+sqrt(2) for example x = 0.1 value = -7.9 sign is -
x = -1+sqrt(2) value = 0
-1+sqrt(2) < x for example x = 10 value = 11.9 sign is +

From this table you can read off the ranges of values for which the expression is >= 0.