Question

Please, help me to solve this: abs(x^2-1)>=abs(2x)-1/3 (algebraically )

sorry

Answer 1

\[\left| x^2-1 \right|\ge \left| 2x \right|-\frac{ 1 }{ 3 }\]

Answer 2

http://fooplot.com/plot/bapz7l4kw9

Answer 3

based on the graph, there are 3 regions where the red line is above the green line
those are your answer

Answer 4

algebraically ?

Answer 5

we would have to break the problem into pieces.

notice that x^2 - 1 is always positive when x^2 ≥ 1, and so is 2x
so for x≥1 we can solve
x^2 -1 ≥ 2x - 1/3
or
x^2 -2x - 2/3 ≥ 0

(x - 1 + sqrt(5/3))( x -1 - sqrt(5/3)) ≥ 0

x≥ 1 + sqrt(5/3) is one answer.

Based on the graph, I think we will find -1 - sqrt(5/3) will be another, but we can check by assuming x ≤ -1 and solving

x^2 -1 ≥ -2x - 1/3

also, check the region between 0 and +1
and the region between -1 and 0

Answer 6

phi, I'll try to study your answer. Thank you for the attention.