maths.. inequalities!

1) | x - 4 | >= 7
If x >= 4, x - 4 >= 7 => x >=11
If x < 4, 4 - x >=7 => -3 >= x
Therefore the solution is x >= 11 or x <= -3
2) | 4x | < x - 2
if x >= 0, 4x < x - 2
3x < - 2
x < -2/3 (rejected)
if x < 0, -4x < x - 2
2 < 5x
2/5 < x (rejected)
There is no solution for this inequality
3) | x - 5| <= | 3 - x |
Case (I) if x - 5 >=0, i.e x >= 5
x - 5 <= x - 3
-5 <= -3 which is always true
Case(II) If 3 < x < 5
5 - x <= x - 3
8 <= 2x
4 <= x
Therefore 4 <= x < 5
Case(III) If x <= 3
5 - x <= 3 - x
5 <= 3 which is impossible
Combining 3 cases, the solution is x >= 4
4) | x - 3 | + | x + 7 | <= 12
Case (I) x <= -7,
3 - x - 7 - x <= 12
-16 <= 2x
x >= -8
Therefore -8 <= x <= -7
Case (II) -7 < x < 3
3 - x + x + 7 <= 12
10 <= 12 which is always true
Case (III), x > 3
x - 3 + x + 7 <= 12
2x + 4 <= 12
x <= 4
Therefore 3 < x <= 4
Combing the 3 cases, -8 <= x <= 4