How can I solve for x in the inequality 1/x <= 4?

[mackhina]

Homework Statement

1/x <= 4

Homework Equations

The Attempt at a Solution

I initially converted 1/x back to x^-1 which gave me the answer x <= 1/4 which makes sense, but I should also get x < 0 which I'm not sure about how to get via solving?

Also is converting 1/x to x^-1 the best method to get this answer? In my notes I read that I should divide be a variable, which makes sense I guess as I don't know if it's positive or not? Any help in explaining a better process of evaluating this problem would be heaps appreciated!

Cheers

Mick

 

[Mark44]

Homework Statement

1/x <= 4

Homework Equations

The Attempt at a Solution

I initially converted 1/x back to x^-1 which gave me the answer x <= 1/4 which makes sense,

Actually, it doesn't make sense. If you look at the graph of y = 1/x, when x <= 1/4, y >= 4, which isn't what you're given.

There is no need to convert 1/x to x^-1, but when you multiply an inequality by a variable, it makes a difference whether the vaariable is positive or negative.

but I should also get x < 0 which I'm not sure about how to get via solving?

Also is converting 1/x to x^-1 the best method to get this answer? In my notes I read that I should divide be a variable, which makes sense I guess as I don't know if it's positive or not? Any help in explaining a better process of evaluating this problem would be heaps appreciated!

Cheers

Mick

 

[mackhina]

You're right, the inequality should be around the other way. I plotted the graph but now I'm even more confused?

What part of the equation is swapping the inequality sign? I didn't multiply or divide by a negative as I left the variable on the same side?

Am I just suppose to do this question through observation?

 

[statdad]

You have two ways (at least) to do this.

First, break into cases:
i) Assume x &gt; 0, and then solve \frac 1 x \le 4 based on this
ii) Assume x &lt; 0, and solve the inequality based on this

Second choice
Write the inequality as
<br /> \frac 1 x - 4 \le 0 \Rightarrow \frac{1-4x} x \le 0 <br />

If you can determine the signs of the numerator and denominator of the second fraction you can determine the solutions.

 

[zgozvrm]

You're right, the inequality should be around the other way. I plotted the graph but now I'm even more confused?

What part of the equation is swapping the inequality sign? I didn't multiply or divide by a negative as I left the variable on the same side?

Am I just suppose to do this question through observation?

Observation is probably the easiest way to see it.

As already stated, there are 2 answers to this problem. The answer you gave (with the inequality reversed) can be algebraically done this way:

\frac{1}{x} \le 4 \rightarrow \frac{1}{x}x \le 4x \rightarrow 1 \le 4x \rightarrow \frac{1}{4} \le \frac{4x}{4} \rightarrow \frac{1}{4} \le x

Which is how the inequality gets "turned around."

 

[Mark44]

Observation is probably the easiest way to see it.

As already stated, there are 2 answers to this problem. The answer you gave (with the inequality reversed) can be algebraically done this way:

\frac{1}{x} \le 4 \rightarrow \frac{1}{x}x \le 4x \rightarrow 1 \le 4x \rightarrow \frac{1}{4} \le \frac{4x}{4} \rightarrow \frac{1}{4} \le x

Above, to get the second inequality, the tacit assumption is that x > 0.

Which is how the inequality gets "turned around."

 

[mackhina]

Thanks everyone for your help, I think I've got it now.

I've gotten that 1/4 <= x by assuming that x>0 I can bring it across

and

x<0 when x is negative.
I had a hard time trying to work out why I couldn't get the answer x=0 mathematically. My understanding of the question now though is that if I substitute any negative number into 1/x it will result in a negative number, which is always less than 4. So I should solve this by reasoning rather than trying to have a number pop out.

Thanks again.

 

[HallsofIvy]

Thanks everyone for your help, I think I've got it now.

I've gotten that 1/4 <= x by assuming that x>0 I can bring it across

and

x<0 when x is negative.
I had a hard time trying to work out why I couldn't get the answer x=0 mathematically.

What do you mean "get the answer x= 0"? x= 0 is NOT a solution to this problem. You should be able to see immediatly that if x= 0, 1/x is not defined.
If x> 0, then you can multiply both sides of 1/x<= 4 by the positive number
get 1<= 4x, then divide both sides by the positive number 4 to get 1/4<= x.
If x< 0, then you can multiply both sides of 1/x<= 4 by the negative number x, reversing the inequality because you are multiplying by an negative number: 1>= 4x. Then divide both sides by the positive number 4 to get 1/4>= x. However, since x< 0, "1/4>= x" is the same as 0> x.

My understanding of the question now though is that if I substitute any negative x to number into 1/x it will result in a negative number, which is always less than 4. So I should solve this by reasoning rather than trying to have a number pop out.

Thanks again.

 

[mackhina]

I see, so the answer should be that 1/4>= x, but because I defined x<0, I move my answer further along the number line until it meets that set criteria. That's easier to understand!

Thanks for helping me out HallsofIvy.

Cheers

Mick