Racking my brain

[Chrispy]

Year 13 New Zealand textbook from 1989.
Solve for all real values of x: 1/x² ⦥ 2/(x+k)²
Answer in book, wondering if it's an error. Thanks for any help or tips.

 

[stapel]

Solve for all real values of x: 1/x² ⦥ 2/(x+k)²

Is the character after the "1/x^2" supposed to be the "greater than or equal to" symbol?

Answer in book, wondering if it's an error.

Why? What did you get when you worked the exercise?

Please reply showing what you did, what answer you got, and the answer from the book that you believe is to be in error. Thank you!

 

[Chrispy]

Thanks for responding guys.

I can't tell you everything I did, it would take hours. But I kept running into this:

k/x ≥ ± √2 -1 (that's a better greater-than-or-equal-to sign isn't it.)

Answer in book is this:

(1-√2)k < x < (1+√2)k

Also don't know why sign changes from non-strict to strict inequality.

Cheers

 

[Chrispy]

And I didn't want to ask, but surely there's a better way to put in the mathematical symbols than what I've been doing which is copying and pasting from other pages? Feel free to roll your eyes.

 

[JeffM]

Year 13 New Zealand textbook from 1989.
Solve for all real values of x: 1/x² ⦥ 2/(x+k)²
Answer in book, wondering if it's an error. Thanks for any help or tips.

\(\displaystyle \dfrac{1}{x^2} \ge \dfrac{2}{(x + k)^2} \implies\)

\(\displaystyle (x + k)^2 * x^2 * \dfrac{1}{x^2} \ge (x + k)^2 * x^2 * \dfrac{2}{(x + k)^2} \implies\)

\(\displaystyle (x + k)^2 \ge 2x^2 \implies x^2 + 2kx + k^2 \ge 2x^2 \implies\)

\(\displaystyle 0 \ge 2x^2 - (x^2 + 2kx + k^2) = x^2 - 2kx - k^2.\)

\(\displaystyle 0 = x^2 - 2kx - k^2 \implies x = \dfrac{- (- 2k) \pm \sqrt{(-2k)^2 - 4(1)(-k^2)}}{2 * 1} = \dfrac{2k \pm \sqrt{8k^2}}{2} = k\left(1 \pm \sqrt{2}\right).\)

With me to here? I am not quite sure what you did from here to get \(\displaystyle \dfrac{k}{x} \ge \pm\ \sqrt{2} - 1.\)

Are there any constraints on k?

 

[JeffM]

And I didn't want to ask, but surely there's a better way to put in the mathematical symbols than what I've been doing which is copying and pasting from other pages? Feel free to roll your eyes.

There is a better way, but it is fussy, and you are trying to learn math, not a formatting language.

We can understand things if you follow PEMDAS and use sensible symbols like >= for \(\displaystyle \ge\).

 

[mmm4444bot]

surely there's a better way to put in the mathematical symbols than ... copying and pasting from other pages

If you're using Windoze, the Character Map is easier.

 

[lookagain]

\(\displaystyle x = \dfrac{- (- 2k) \pm \sqrt{(-2k)^2 - 4(1)(-k^2)}}{2 * 1} = \dfrac{2k \pm \sqrt{8k^2}}{2} = k\left(1 \pm \sqrt{2}\right).\)

Are there any constraints on k?

\(\displaystyle x \ = \ \dfrac{2k \pm \sqrt{8k^2}}{2} \ \implies \)


\(\displaystyle x \ = \ \dfrac{2k \pm 2|k|\sqrt{2}}{2} \ \implies \)


\(\displaystyle x \ = \ k \pm |k|\sqrt{2} \)


If k is non-negative, then the above step leads to


\(\displaystyle x \ = \ k \pm k \sqrt{2}\ \implies\)


\(\displaystyle x \ = k(1 \pm \sqrt{2})\)


- -- - - - - -- - - - - - -- - - - - - - - - - --- - - - -- - - - -- - - - - -

Answer in book is this:

(1-√2)k < x < (1+√2)k

Chrispy, for one, that can't be the answer, because it includes x = 0, but x = 0 is

restricted from the original equation.

 

[Dale10101]

Mathtype

And I didn't want to ask, but surely there's a better way to put in the mathematical symbols than what I've been doing which is copying and pasting from other pages? Feel free to roll your eyes.

I think there are Latex editor options that are free if you search the web including online editors but I have not looked into them.

If you find a useful simple freebie maybe you can post your find.

 

[JeffM]

I think there are Latex editor options that are free if you search the web including online editors but I have not looked into them.

If you find a useful simple freebie maybe you can post your find.

Dale

There is a LateX editor incorporated into the site. It's just a pain for students to fuss with when they are trying to focus on math.

 

[Dale10101]

OK

Dale

There is a LateX editor incorporated into the site. It's just a pain for students to fuss with when they are trying to focus on math.

Gotcha, I use a commercial program which facilitates writing equations and then translates to Latex if desired but even that is still no where as simple as pencil, paper, eraser and using the printer to scan a copy into a .jpg or the like for posting as a pic.

I have played with tablets and scribes and it sort of nice but then you find yourself getting stuck trying to get the right result here and there and so it too gets frustrating. Good old pencil and paper ... and eraser, oh, and legible handwriting.

 

[Chrispy]

No restraints on x or k, except they're real numbers.

Thanks guys, that was a great help.

 

[lookagain]

No restraints on x or k, except they're real numbers. \(\displaystyle \ \) except that in post #9 I pointed out that x cannot equal 0,
so that your book's answer cannot be correct, as it incudes x = 0.

.