1) ` (x-1)^2 < x ( x +3) `
` x^2 - 2x + 11< x^2 + 3x`
` x^2 - 2x + 1 - x^2 - 3x < 0`
` -5 x < -1 `
` x > 1/5`
2) ` 2(x+2)^2 < 2x( x+ 2) + 4`
` 2(x^2 + 4x + 4) < 2x^2 + 4x + 4`
`2x^2 + 8x + 8 < 2x^2 + 4x + 4`
3)
`(x-2) ( x+2) > x ( x-4) `
` x^2 - 4 > x^2 - 4x`
` x^2 -4 - x^2 + 4x > 0`
`4x - 4 > 0`
`4x > 4`
` x > 1`
4)
` 6x^2 - 36 ≥ 6x ( x-2 ) - 5 ( 2x + 1) `
` 6x^2 - 36 ≥ 6x^2 - 12x - 10x - 5`
` 6x^2 - 36 - 6x^2 + 22x + 5 ≥ 0`
` 22x - 31 ≥ 0`
` 22x - 31≥ 0`
` x ≥ 31/22 `
5)
` ( x + 3) ( x-1) < ( x+1)^2 - 4`
` x^2 + 2x - 3 < x^2 + 2x + 1 -4 `
` x^2 + 2x - 3 - x^2 - 2x - 1 + 4 < 0`
0 < 0 ( vô lý)
6)
`( x + 5)^2 - 6 > x( x-5) - ( 3x-7)`
`x^2 + 10x + 25 - 6 > x^2 + 8x - 7 > 0`
`18x + 12 > 0`
`18x > -12`
`x > -2/3`
7)
` ( x+ 3) ( x^2 - 3x + 9) - 2x ≥ x^2 - 7 `
` x^3 + 27 - 2x ≥ x^2 - 7`
` x^3 - x^2 + 2x + 34 ≥ 0`
` ( x-1) ( x^2 - 1) + 34 ≥ 0`
` ( x-1) ( x-1) (x+1) + 34 ≥ 0`
Vì ` ( x-1) ( x-1) ( x+ 1) ` luôn dương với mọi x nên bất phương trình luôn đúng.
8)
` ( x-2)^2 + 6x^2 ≥ x^2 + 7(2x -1)`
` x^2 - 4x + 4 + 6x^2 ≥ x^2 + 14x - 7`
` x^2 - 4x + 4 + 6x^2 - x^2 - 14x + 7 ≥ 0`
` 5x^2 - 18x + 11 ≥ 0`
` ( 5x - 11) ( x-1) ≥ 0`
` 11/5 ≤ x ≤ 1`
9)
` ( 4x + 3)^2 - 2 < ( 4x - 3) ^2 - ( 5x + 4) `
` 16x^2 + 24x + 9 - 2 < 16x^2 - 24x + 9 - 5x - 4 `
` 16x^2 + 24x + 7 < 16x^2 - 29x + 5 `
` 16x^2 + 24x + 7 - 16x^2 + 29x - 5 < 0 `
` 53x - 2 < 0 `
` 53x < 2 `
` x < 2/53 `
10)
\(3 ( x -2)^2 + 9x > 12 + 3 ( x^2 - x + 3)\)
` 3(x^2 - 4x + 4) + 9x > 12 + 3x^2 - 3x + 9 `
` 3x^2 - 12x + 12 + 9x > 12 + 3x^2 - 3x + 9 `
` 3x^2 - 12x + 12 + 9x - 12 - 3x^2 + 3x - 9 > 0 `
` -x > 0 `
` x < 0 `
