Note: I have a=had to edit this a number of times becasue the LaTex compiler is playing up.

It keeps deleting things.

\(\frac{\left(\sqrt{1-4x^2}-1\right)}{x}\ge -4\)

I will look at the domain restrictions first.

I am assuming that you want only real answers.

\(1-4x^2\ge0\\ 1\ge4x^2\\ 4x^2\le1\\ (2x)^2\le1\\ -1\le2x\le1\\ -0.5\le x\le0.5 \)

Oh, I canot divide by zero either so x cannot be zero

\(-0.5\le x<0 \qquad \)

or \(0< x<0.5\)

Now that I have an initial restriction for x values I will look at the rest.

You cannot just multiply by x as Coolstuff has done because you do not know if x is positive or negative and if it is negative the sign has to be turned around.

Normally the easiest way to handle these is to multiply through by x^2 because we know that x^2 has to be positive. But in this case that will work well.

SO

When is

\(\sqrt{1-4x^2}-1> 0\\ \sqrt{1-4x^2}>1\\ 1-4x^2>1\\ -4x^2>0\\ NO\;\;solutions.\)

If I had thought about it before hand I could have seen that this was the case.

So

\(\sqrt{1-4x^2}-1\le0\qquad \text{for all values of x}\)

If x is negative then the LHS of the original inequality will always be positive so the LHS will always be bigger than -4

So all values of x where \(-0.5\le x < 0\) will make the equation true.

--------------------------------

If x is positive then

\(\frac{\left(\sqrt{1-4x^2}-1\right)}{x}\ge -4\\ \sqrt{1-4x^2}-1\ge -4x\\ \sqrt{1-4x^2}\ge -4x+1\\ \qquad If\;\;-4x+1<0\quad \text{then this must be true}\\ \qquad If\;\;-4x<-1\quad \text{then this must be true}\\ \qquad If\;\;x>-0.25\quad \text{then this must be true}\\ \qquad \text{Since I am only looking at positive values of x, x must be greater than -0.25}\)

So the solution set is [-0.5, 0) union (0, 0.5]