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 &gt; 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&lt;0, pick, say, -1. Clearly 1/-1 - 4 &lt; 0.

For 0 &lt; x &lt; 1/4, pick, say, 1/8. Then 1/(1/8) - 4 = 8 - 4 &gt; 0.

For x &gt; 1/4, pick 1. 1/1 - 4 &lt; 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.