Solving Inequalities

Question

16x-x³≥0

Answer 1

16x - x³ ≥ 0

 

x(16 - x²) ≥ 0

 

x(4 - x)(4 + x) ≥ 0

 

x(4-x)(4+x) = 0 when x = 0, 4, -4

 

Plot these values on the number line:

 

_______-4______0_____4_______

 

 

When x < -4, x and 4+x are negative and 4-x is positive, so                                             x(4-x)(4+x) > 0

 

When -4<x<0, x is negative and the other two factors are positive, so                               x(4-x)(4+x) < 0

 

When 0<x<4, all three factors are positive, so         x(4-x)(4+x) > 0

 

When x > 4, 4-x is negative and the other two factors are positive, so                                  x(4-x)(4+x) < 0

 

Solution:  (-∞, -4] ∪ [0, 4]