This is not an inequality...
Hello there!
I have been struggling with the next equality for some time now. The problem is, that I can get to the correct answers, but they are in the wrong intervals, and I can not figure out, what is it, that I am missing.
So:
| (3x-2)/(x-1) | = 2
I understand, that the root is 2/3, and that if x>= 2/3, the expression inside the absolute value sign is positive, and negative otherwise. After taking the different cases into consideration, I get that if x is the element of (-inf,2/3), the solution is 4/5, and if x is the element of [2/3,inf), the solution is 0. I simply don't know, what am I doing wrong, I have had no problems, solving absolute value equations before, but this rational one confuses me a bit.
Clearly, (though maybe not) this is an inequality, since if it is an equality, the solutions are 0, 4/5.
Assuming this is
you require the fraction to be >2 or <-2
when we have +/+ or -/-
when we have +/- or -/+
x>1 and 3x>2 for +/+
x<1 and 3x<2 for -/-
x>1 and 3x<2 for -/+
contradiction as x>1
x<1 and 3x>2 for +/-
All in all, you have x>0.8 and x<0
(sorry, my modem was jammed and saw no responses)
Hello, szucslaszlo!
. .
I blush to confess that I got tangled up in my algebra,
. . so I tried a graphical approach.
The graph of looks like this:
Code:| | : | :* | : | : * | : * | : * | : * 3| : * - - - + - - - - : - - - - - - - - * | : * : | * : --------+----*----:------------------ | * :1 | : | * : | : | : | *: | : | : | : | *: |
In the graph of
. . everything below the -axis is reflected upward.
Code:| | : | :* | *: | : * | : * | : * | *: * 3| : * - - - + - - - - : - - - - - - - - * | * : * : | * * : --------+----*----:------------------ | :1 | : |
And we want the points where
Code:| | : | :* | : | : * | : * | : * | *: * | : * . . . + . . . . : . . . . . . . . * | *: - - - ♥ - - - ♥ : - - - - - y = 2 | * * : -------+----*----:------------------ | :1 | : |
Now that I can see the solution, I can do the algebra.
The answers are: .