2011/100) Solve for x: (2x + 14) / (5x - 1) ≤ x + 3.

we need to get rid of denominator so multiplying by (5x-1) is problematic as we do not know whether it is positive or negative

so multiply by (5x-1)^2 which is positive to get

(2x+ 14) (5x -1 ) <= (5x-1)^2(x+3)

or (2x+14) (5x-1) - (5x-1)^2 (x+ 3) <= 0

or (5x-1) ((2x+14 - (5x- 1 )(x+3)) < = 0

or (5x-1)(2x+14 - (5x^2 + 14x - 3)) <= 0
or (5x-1)(-5x^2 - 12x + 17) < = 0
or (5x-1)(5x^2+ 12x - 17) >= 0
or (5x-1) (5x+17)(x-1) > = 0

there are 4 ranges - infinite to -17/5
- 17/5 to 1/5 (in this it is positive but x cannot be 1/5 as 5x-1 is in denominator)
1/5 to 1
and 1 to infinite (in this it is positive

x belongs to [- 17/5, 1/5) U [1, ∞).