Ok, my algebra teacher is bugging me. I’m taking a refresher course in college and we came across this problem:
|x-2|>3
to which I gave the answer:
-1>x>5
she says NO! the actual answer is |x|>5
Now, while her answer certainly works, it’s missing serveral numbers from it’s solution set.
And I asked her how it is even POSSIBLE to just take part of an absolute value equation out and move it around without changing the nature of the problem. She says she will bring me the mathmatical solution next Wednesday. I, however, cannot wait that long for this is driving me nuts! Am I right here?
Yeah, she is missing some numbers from the solution set. If x < -1 or x > 5, the inequality is satisfied. btw, you can’t have -1 > x > 5, because that implies that -1 > 5.
If you’re solving an inequality of the form |f(x)| > a (or <) for some positive number a, what you have to do is break it up into two cases: f(x) >= 0, or f(x) < 0, since the absolute value function behaves differently depending on whether f(x) is negative or not.
Ultimately this comes down to:
|f(x)| > a
is broken up into:
f(x) < -a, or f(x) > a.
|f(x)| < a
becomes
-a < f(x) < a.
A possibly helpful hint for the future. I think it’s often helpful to think of |x-y| as expressing the distance between x and y. So your problem becomes, “What numbers are more than three units away from 2?” So anything to the left of 2-3 = -1 is a solution, and anything to the right of 2+3 = 5 is also a solution.
It’s really scary that your teacher would say that. This is really basic stuff.
The most basic rule is:
If |x|>k then (-k > x > k)
Which of course means that
If |x-2|>3 then (-3 > x-2 > 3)
For the good of mankind you should report her to the police.
On the other hand for the good of Enore_Tsotset, you should lay low and never bring the subject up again.
I, of course, would make a citizens arrest, but only in a disguise, and under an assumed name.
There seems to be a little confusion here regarding removing the absolute value signs. If you have |x| < k, then -k < x < k. However, if you have |x| > k, then you have x < -k or x > k. Saying -k > x > k implies that both are true at the same time, which is impossible for k > 0.
As somebody pointed out earlier, -k > x > k is not a legitimate way to write the result. What this means, by standard mathematical convention, is that -k > x and x > k, and this in turn implies that -k > k, which of course is not true unless k is negative. What you are trying to say is -k > x or x > k, and there is no convention for combining these into one statement.
No, it isn’t surprising that a HS math teacher would make this mistake. In my experience, the math ability of many HS math teachers is much lower than you think and even many of the ones that are able to not make many mistakes still only know the routine and not have a overall view of the field.
When I used to teach summer courses to HS Math teachers, I tended to see them as very good Freshman college math students, average Sophomore or poor Junior/Senior.
There are many exceptions of course but the range of ability between them was truely scary to me.
That’s what scares me, this IS college. And a reputable one at that. I’ve been taking night school for a year and this is my first math course. I remember most of this from HS algebra so I’m ok in the class, but the other students are lost, and she’s teaching them to do it (acording to my text book and the posters in this thred) the wrong way! Oh well, just do what she says is right and move on…
I wouldn’t recommend doing that, necessarily. A teacher teaching incorrect methods is worse than no teacher at all. The nice thing about mathematics is that you can prove to her that she’s wrong, if that’s what she’s doing.
For example, ask her to solve |x-5| < -4 using her method.
On one hand (the correct hand), it should be obvious there’s no solution.
On the other hand (the incorrect, using-her-method hand), the solution is |x| < 1.
Obviously something’s wrong, and, also obviously, it’s her method of solving the inequality that’s wrong. Don’t let up until you convince her of this.
Well, when questioned vigorously by me last night, she said she was going to bring me a proof next Wednesday for why her method is correct. I’m interested to see what this is. She claims its from a teaching guide she has…
Enore, you may not want to make waves and that’s fine but there is a good chance that your teacher (who I didn’t know was COLLEGE btw) is part time and there is someone that will oversee these people. I used to have that job as well as teaching duties. This person should be informed, even in secret.
If your teacher is a full time prof and makes mistakes like this regularly (everyone brain farts every now and then) then that is really bad. There is nothing more pathetic than regularly making bonehead mistakes in your chosen field. Sadly, I saw that happen many times when I was in academia. I guess that’s what happens when you pay so little for these positions – you get lower quality people applying.
If she doesn’t correct her mistakes soon, find this person that oversees these people.
*Enore_Tsotset *, a college teacher who couldn’t see the wrongness of her answer when you brought it to her attention isn’t going to be much help to you in understanding algebra. If there are any other sections available, I’d recommend that you switch pronto.
Cabbage has a good example for her to solve. Ask her to solve it!
I don’t understand what is so flipping hard about step functions (expecially 2-step like absolute value) and logarithms but people really, really seem to have trouble with them. I think it has to do with people knowing HOW to solve something by following a procedure instead of learning WHY the procedure works. My students seemed to have little problems with absolute value and logs because I drilled the WHY into them, I guess.
Knowing HOW to solve something by knowing a procedure is trivial and it is NOT the same as knowing what you are doing! That knowledge will dissappear 5 minutes after the final exam. Knowing WHY you are doing the steps you are doing is where real understanding is.
She just called me at work today to tell me that I was absolutly right and that she had printed out the wrong test. I’m still confused how she didn’t see my point, though, considering I had written out several examples on the board. She thought the test she handed out did not have the absolute value brackets and I was arguing the wrong point. I need out of her class… Thanks for the help everyone!
I need out of her class… Thanks for the help everyone!