3題 數(inequalities) (1)

1. 6x^2 + 5x + 1 > 0

(2x + 1)(3x + 1) > 0

{2x + 1 > 0
{3x + 1 > 0

or
{2x + 1 < 0
{3x + 1 < 0

{x > -1/2, x > -1/3 or {x < -1/2, x < -1/3

Therefore, we have x < -1/2 or x > -1/3



2. 4x^2 - 4x + 3 < 0

(2x - 3)(2x + 1) < 0

{2x - 3 < 0
{2x + 1 > 0

or
{2x - 3 > 0
{2x + 1 < 0

{x < 3/2, x > -1/2 or {x > 3/2, x < -1/2 (no solution)

Hence, we have -1/2 < x < 3/2.


3. (x + 1)^2 > 3x - 5

x^2 + 2x + 1 > 3x - 5

x^2 - x + 6 > 0

(x - 3)(x + 2) > 0

x < -2 or x > 3