Simple Inequality with Modulus Question

[kuskus94]

Homework Statement

Determine m :

|x-10|<{1}/{m}

if its final form is :

|x^{2}+{4}x-140|<1

Homework Equations

To remove the modulus, square them...

The Attempt at a Solution

I have tried to assume that if

|x-10|{m}<{1}

then, I can find

|x-10|{m}=|x^{2}+{4}x-140|

|x-10|{m}=|(x+14)(x-10)|

but, can I omit (x-10)?

 

[haruspex]

|x-10|{m}<{1}

You can't do that because m might be negative for some x. Try the squaring hint.

|x-10|{m}=|x^{2}+{4}x-140|
|x-10|{m}=|(x+4)(x-10)|

There's an error in that step.

 

[kuskus94]

You can't do that because m might be negative for some x. Try the squaring hint.

There's an error in that step.

Okay, sorry it should be

|x-10|{m}=|(x+14)(x-10)|

So, if I do this :

(x-10)^2{m}^2=((x+14)(x-10))^2

is it possible?

 

[haruspex]

it should be

|x-10|{m}=|(x+14)(x-10)|

Not quite. Should be |x-10||m|=|(x+14)(x-10)|
But if you go back to |x−10|<1/m you can observe that m is necessarily non-negative, which simplifies the logic.

(x-10)^2{m}^2=((x+14)(x-10))^2

So what form should m take to make that guaranteed, and ensure m non-negative?

 

[kuskus94]

Not quite. Should be |x-10||m|=|(x+14)(x-10)|
But if you go back to |x−10|<1/m you can observe that m is necessarily non-negative, which simplifies the logic.

So what form should m take to make that guaranteed, and ensure m non-negative?

because |m|=\frac{|(x+14)(x-10)|}{|(x-10)|}, is it |m|=|(x+14)| ?

 

[haruspex]

because |m|=\frac{|(x+14)(x-10)|}{|(x-10)|}, is it |m|=|(x+14)| ?

Almost. If m were negative, would it satisfy the requirements?

 

[kuskus94]

Almost. If m were negative, would it satisfy the requirements?

Hmmm... so, I need to add m&gt;0 to the equation or something like that.

m=|x+14|; m&gt;0?

 

[haruspex]

Hmmm... so, I need to add m&gt;0 to the equation or something like that.

m=|x+14|; m&gt;0?

Yes, except that you don't now need to specify m >= 0. It follows from m=|x+14|.

 

[kuskus94]

Yes, except that you don't now need to specify m >= 0. It follows from m=|x+14|.

So, the final answer is m=|x+14|?

I am still in doubt.

 

[haruspex]

So, the final answer is m=|x+14|?

I am still in doubt.

Looks right to me. Did you try plugging it into the original inequality?