# Solutions of an inequality (1-√(1-4x^2)/x < 3

• MHB
• Vali
In summary, the conversation discusses finding the solutions to the inequality (1-sqrt(1-4x^2)/x < 3, with the conditions x≠0 and 1-4x^2≥0. The correct answer is [-1/2,0)U(0,1/2], but there is confusion about the process used to find the solutions. The expert suggests solving the associated equation (1- sqrt(1- 4x^2))/x= 3 to handle the inequality, and explains that the associated equation should actually be (1- sqrt(1- 4x^2))/x= 3. The conversation ends with clarification about the roots of the quadratic equation
Vali
I need to find the solutions of the following inequation:
(1-sqrt(1-4x^2)/x < 3
I put the conditions x different from 0 and 1-4x^2>=0 and I got [-1/2,0)U(0,1/2] which is the right answer but I'm confuse because I usually subtract 3 to get (1-sqrt(1-4x^2)/x - 3 < 0 then, after I made some work and I got a fraction, I find the variation of each function (from numerator and denominator) and in the final I find the sign of f(x) which should be negative in our case and like this I find the solutions, but I didn;t get the same result.

Vali said:
I need to find the solutions of the following inequation:
(1-sqrt(1-4x^2)/x < 3
I put the conditions x different from 0 and 1-4x^2>=0 and I got [-1/2,0)U(0,1/2] which is the right answer but I'm confuse because I usually subtract 3 to get (1-sqrt(1-4x^2)/x - 3 < 0 then, after I made some work and I got a fraction, I find the variation of each function (from numerator and denominator) and in the final I find the sign of f(x) which should be negative in our case and like this I find the solutions, but I didn;t get the same result.
You are missing a parenthesis. Do you mean (1- sqrt(1- 4x^2))/x< 3?

First, because of the $$\sqrt{1- 4x^2}$$, x must lie between -1/2 and 1/2.

In my opinion the best way to handle such an inequality is to first solve the associated equation, $$\frac{\sqrt{1- 4x^2}}{x}= 3$$. To solve that multiply both sides by x to get $$\sqrt{1- 4x^2}= 3x$$ and square both sides to get $$1- 4x^2= 9x^2$$. Then $$13x^2= 1$$, $$x= \pm\sqrt{1/13}$$.

The point of that is that in order that the value of a function change from "< 3" to "> 3" it must either be "= 3" or be discontinuous. This function is 3 at $$x=\pm\sqrt{1/13}$$ and, because of the division by 3, is discontinuous at x= 0.

Now, check one value in each interval, $$-1/2\le x\le -\sqrt{1/13}$$, $$-\sqrt{1/13}\le x< 0$$, $$0< x\le \sqrt{1/13}$$, and $$\sqrt{1/13}\le x\le 1/2$$ to determine in which interval the value is less than 3 and in which the value greater than 3.

Thank you for the help!
I understood, but I have one question.Why the associated equation is sqrt(1-4x^2)/x=3 and not (1-sqrt(1-4x^2))/x=3 ? Why you cancel that "1" ?

I just mistakenly dropped it! Yes, the associated equation should be $$\frac{1- \sqrt{1- x^2}}{x}= 3$$ so that $$1- \sqrt{1- 4x^2}= 3x$$. Then $$\sqrt{1- 4x^2}= 1- 3x$$ and, squaring both sides, $$1- 4x^2= 9x^2- 6x+ 1$$. That gives us the quadratic equation $$13x^2- 6x= x(13x- 6)= 0$$ which has roots 0 and -6/13.

Country Boy said:
That gives us the quadratic equation $$13x^2- 6x= x(13x- 6)= 0$$
which has roots 0 and -6/13.
0 and 6/13 :)

## 1. What is an inequality?

An inequality is a mathematical statement that compares two quantities and shows how they are not equal. It uses symbols such as <, >, ≤, and ≥ to represent the relationship between the quantities.

## 2. How do you solve an inequality?

To solve an inequality, you need to isolate the variable on one side of the inequality symbol. This is done by using inverse operations, just like in solving equations. However, when multiplying or dividing by a negative number, the direction of the inequality symbol must be flipped.

## 3. What is the difference between an inequality and an equation?

An inequality compares two quantities and shows how they are not equal, while an equation shows that two quantities are equal. Inequalities use symbols such as <, >, ≤, and ≥, while equations use an equal sign (=).

## 4. What does the solution of an inequality represent?

The solution of an inequality represents all the possible values that make the inequality true. This can be represented on a number line or as an interval of values.

## 5. How do you graph a solution of an inequality?

To graph a solution of an inequality, first solve the inequality to find the boundary points. Then, depending on the inequality symbol, either use a solid line (≤ or ≥) or a dashed line (< or >) to connect the boundary points. Finally, shade the region that includes all the solutions of the inequality.

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