Solving an Inequality with X in a Denominator in Terms of Intervals

[EcKoh]

I have been tasked with solving the following inequality:

$\frac{1}{x}$ < 4

Attached to this thread is my attempted solution. As you can see I begin with simply solving the inequality for x, and I obtain the result x > $\frac{1}{4}$

Next, I convert the equation into what I thought was the proper form for a hyperbola. I realize now I should have left the equation alone because it was already in proper form. However, I figure now that graphing at this point in my attempt may have not been the correct thing to do.

Next I find the roots for the inequality. I find these to be 0, and $\frac{1}{4}$.

Once the roots are found, I find the possible intervals for the inequality. The intervals I use are the following: x<0, 0<x<$\frac{1}{4}$, and x>$\frac{1}{4}$.

I then set these up on a chart in order to find which intervals solve the inequality. However, I must have either set this up wrong or am going about this the wrong way. Any tips or guidance on where to go from here would be greatly appreciated.

 

[mathman]

I don't understand you question. However 1/x < 4 has solution in two ranges. For x > 0, then x > 4. For x < 0, all x.

 

[EcKoh]

My textbook states that the solution for this problem to be (-∞, 0) $\cup$ ($\frac{1}{4}$, ∞) (meaning that the roots are 0, and 1/4. I just don't know how to arrive at that answer. Basically I am wondering how to arrive at this solution, because I keep working the problem and getting different answers.

 

[Muphrid]

I don't see what the problem is. Your graph on the right clearly shows that for $x > 1/4$, the value of $1/x - 4$ is less than 0 as required.

 

[EcKoh]

So is graphing it the only way to solve the inequality? Or is there a way to do it arithmetically.

 

[Muphrid]

You knew the function must have roots at 0 and 1/4. These are the only points where it can change positive or negative. All you have to do is plug in one value from each region.

For $x<0$, pick, say, -1. Clearly $1/-1 - 4 < 0$.

For $0 < x < 1/4$, pick, say, $1/8$. Then $1/(1/8) - 4 = 8 - 4 > 0$.

For $x > 1/4$, pick 1. $1/1 - 4 < 0$.

 

[EcKoh]

Ah thanks, it turns out I was reaching incorrect solutions because in my notebook I was trying to find for >0 instead of <0...

Thank you very much for your help and for pointing this out for me when I read your last post.

 

[mathman]

I can't appreciate why you have a problem for x < 0. If x < 0, then 1/x < 0, so 1/x < 4.

 

[EcKoh]

That's ok.