How do solve 3/(x-2)<=3/(x+3) algebraically?

1 Answer
Oct 5, 2016

-3<x<2

Explanation:

Slower way:

  • Bring everything to the right: 3/(x-2) - 3/(x+3) \leq 0

  • Lowest common denominator: (3(x+3)-3(x-2))/((x-2)(x+3))\leq 0

  • Expand the numerator: (3x+9-3x+6)/((x-2)(x+3))\leq 0

  • (9+6)/((x-2)(x+3))\leq 0

  • (15)/((x-2)(x+3))\leq 0

  • Since 15 is always positive, the sign of the fraction is decided by the sign of the denominator. (x-2)(x+3) represents a parabola with zeros in x=-3 and x=2. Such a parabola is negative between its solutions, as you can see in the graph here:
    graph{(x-3)(x+2) [-4.96, 6.146, -2.933, 2.614]}