Need some help with this inequality please?!?

Question

x - 1
------ ≥ 0
x + 2

Thanks...and the greater than or equal to symbol code is
& g e ;
with no spaces.

Answer 1

Multiply by x+2, realizing that if it is negative, it flips the inequality:

case 1: x+2 is positive

(x - 1)/(x + 2) >= 0
(x - 1) >= 0 (when x+2 is positive)

x >= 1, AND x+2 is positive (i.e., x+2>0, x>-2)

That's x >= 1, the more restrictive of the two combined ranges.

case 2: x+2 is negative

(x - 1)/(x + 2) >= 0
(x - 1) <= 0 (when x+2 is negative)
x <= 1 AND x+2 is negative (i.e., x+2<0 x<-2)

That's x < -2, the more restrictive of the two combined ranges.

So the overall answer is two ranges:

(a) x >= 1
(b) x < -2

================
Let's check with some numbers.

x = 2 is in the x>=1 part of the solution:
(x - 1)/(x + 2) = (2-1)/(2+2) = 1/4, which is greater than or equal to zero

x = 0 is NOT in the solution:
(x - 1)/(x + 2) = (0-1)/(0+2) = -1/2, which is NOT greater than or equal to zero

x = -3 is in the x<-2 part of the solution:
(x - 1)/(x + 2) = (-3-1)/(-3+2) = -4/-1 = 4, which is greater than or equal to zero

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Note: there is a shortcut that can be used in this case, involving the sign of the numerator and denominator, but that only works because the inequality involves zero.

The procedure I outline above will work if the equation had instead been >= 1, or any other number.

Answer 2

(x-1)/(x+2) &ge 0

a fraction is positive iff the numerator and denominator have the same sign. so two possibilities

x-1 &ge 0 AND x+2 > 0 or x &ge 1
x-1 &le 0 AND x+2 < 0 or x < -2

so x < -2 OR x &ge 1