# Find all the value of `x` satixfying the condition in as many way as possible: `root(3)(25x(2x^2+9))>= 4x+3/x` ` ` ` `

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### 1 Answer

We are asked to find all values for x satisfying `root(3)(25x(2x^2+9))>=4x+3/x`

(1) Looking at the graph of the left side and the right side it appears that the condition holds for all x<0. The graph of the left side in black, the right side in red:

But you might see a point of equality in the first quadrant.

(2) `root(3)(25x(2x^2+9))>=4x+3/x` Cube both sides -- this does not change the inequality

`50x^3+225x>=64x^3+144x+108/x+27/(x^3)`

Multiply by `x^3` -- note that we have two cases:

(a) x>0 ; the inequality remains the same.

`50x^6+225x^4>=64x^6+144x^4+108x^2+27`

`14x^6-81x^4+108x^2+24<=0`

This function is nonnegative for all x>0 -- however it is zero at `x=+-sqrt(3)` .

Thus in the original inequality, there is an equality at `x=sqrt(3)`

(b) x<0; the inequality changes.

`14x^6-81x^4+108x^2+24>=0` is true for all x<0.

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The solutions to `root(3)(25x(2x^2+9))>=4x+3/x` are x<0 and `x=sqrt(3)` . (There is equality at `x=+-sqrt(3)` )

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