MHB Interval Notation of Inequality: -9<1/x<=1

[Colin2]

I need help determining the interval notation of the inequality below:

-9<1/x<=1

 

[MarkFL]

If we are told that:

$$a<b$$

then doesn't this imply:

$$\frac{1}{a}>\frac{1}{b}$$?

 

[Colin2]

So would I be right If I concluded that the interval is?:

(-9,-1] U (0,1]

Since 0 can't be included as we are working with 1/x?

 

[MarkFL]

So would I be right If I concluded that the interval is?:

(-9,-1] U (0,1]

Since 0 can't be included as we are working with 1/x?

No, what I would do is take the original inequality:

$$-9<\frac{1}{x}\le1$$

And break it up into the system:

$$-9<\frac{1}{x}\tag{1}$$

$$\frac{1}{x}\le1\tag{2}$$

Now, can you rewrite this system by inverting both sides of both inequalities?

 

[Colin2]

So...

-1/9<x & x<=1?

 

[MarkFL]

So...

-1/9<x & x<=1?

No, look at my first reply...when we invert both sides of an inequality, we must change its direction, and so we would get:

$$-\frac{1}{9}>x\tag{1}$$

$$x\ge1\tag{2}$$

Now, how would you write the solution in interval notation?

 

[Colin2]

Ohhh yeah I forgot about that, also I'm new here so I'm not familiar with the way people express maths symbols and stuff using code so I'm just going to write it an easier way.

How about:

(infinity,-1/9) U [1,infinity)?

 

[MarkFL]

Ohhh yeah I forgot about that, also I'm new here so I'm not familiar with the way people express maths symbols and stuff using code so I'm just going to write it an easier way.

How about:

(infinity,-1/9) U [1,infinity)?

Yes...I think you simply forgot to put the negative sign before $\infty$ on the left...

$$\left(-\infty,-\frac{1}{9}\right)\,\cup\,\left[1,\infty\right)$$

 

[Colin2]

Yeah because I wrote it in text...thanks for the help Mark!

Also could you point me in the direction of how to represent maths symbols on this forum?

 

[MarkFL]

Yeah because I wrote it in text...thanks for the help Mark!

Also could you point me in the direction of how to represent maths symbols on this forum?

When you are composing a post, to include $\LaTeX$, one of the easiest ways is to first click the $\Sigma$ button in the editor toolbar directly above the text area. Clicking this button will generate the $$$$ tags, and your cursor will be located between the tags. You may then begin entering the code.

If you look to the right of the editor you will see our "Quick $\LaTeX$" element which you can use to generate many commonly used symbols and commands. We have these divided into categories which you can access using the drop-down menu at the top.

If you look below the editor, you will see our $\LaTeX$ Live Preview where you can test out your $\LaTeX$ quickly to make certain you have it like you want, and then you can copy what's there to the editor. The Quick $\LaTeX$ element works there as well. :D

 

[Colin2]

Thank you very much again for the help!