Hi
I've gone through the "Solving inequalities" article. However, I didn't find anything in the document or my textbooks that could help me solve this inequality:
[(|x|-3)/ (x-2)] < 2x
Could someone please guide me in solving this?
Thanks in advance.
You need to consider a few cases, since when you multiply by a negative number, the inequality sign changes direction.
.
First, it should be clear that .
Case 1: , then
.
Since and is therefore positive, we can say , so
or
or .
Since we know that , we can say satisfies the original inequality.
Case 2: , then
.
Since , we need to consider that if and if .
Case 2a) , then
.
Since we know , that means satisfies the inequality.
Case 2b) , then
.
Since the LHS is always positive, this inequality is never true.
So putting it all together, the solution to the inequality
is
.
Thanks for the help.
I'm sorry that I stated that I was looking for the solution to this inequality. What we were supposed to find out was the range of x.
However the answer to this is (3-33^0.5)/4 < x < 2 or x < 3.
There's one more thing that I find different from your approaches. In the text, it's stated that to remove the denominator from either the LHS or RHS, we have to multiply the square of the denominator to both sides. However, the steps shown above just require me to bring the denominator over to the other side.
Are both these methods one and the same?
Alternatively...
The 3 cases are
(1)
Both factors must be positive or both must be negative while
Or
.... impossible
(2)
We will be multiplying both sides by a negative value, so we reverse the inequality sign.
... impossible
... a valid solution.
(3)
For this quadratic, and it's roots are complex, since they are
The solution is