
Inequalities
First part of the question says: Solve the inequality
And after solving, the answer is
Then the next part of the question says: Hence deduce the solution to the inequality
On comparing, i replaced x in the answer to the first part with
so,
and
and
therefore,
Yes, that's right. The answer is correct but shouldn't the answer also be dealt with?
So shouldn't the right answer be... ? why is this not the answer? Please advise, thanks

The difficulty is that your solution to the original inequality is wrong. There is no reason to say . It is that would cause the left side to not exist.

sorry that was a typo, so why shouldn't the solution include

is never negative so there no need to exclude 1/3. And, if x= 1/9, then [tex]\sqrt{x}= 1/3[/itex], not 1/3.

Problem 1
Let's change the equation up a bit to really look at it. All we'll do is pull a 1 out of the to give , then rearrange it:
So this is what we already know by looking at it:
1. We have a negative degree two polynomial expression on top
2. This expression has two real 0s so at some point x will be > 0
2. We can easily figure out our zeros now for the equation (1 and 4)
3. Also it looks like if because we cannot divide by 0, there would be no solution.
So our answer is:
and
Note another way to write this solution:
and
Problem 2
Now on your next problem:
It's going to be mostly the same except for two differences:
1. Solving for our roots won't be as easy, for example:
There is no solution here, the sqrt(x) will never be negative.
No solution, so only 16 for a root.
So that was a little different but not too bad.
2. There will never be a real value for x that would create a 0 in the bottom of your expression on the right
For real numbers in x:
So we don't have to worry about a "no solution" problem on the bottom.
And we understand that sqrt{x} will always remain nonnegative. To explain this better, if we put a negative number in for x, we get a complex number in return. If we put a 0 in for x, we get a 0. And if we put a positive number in for x, we get a positive in return. Never a negative result.
We can form our answer now to be
Or in other words, because is nonnegative:
And thusly we have our solution:
Good luck you were on the right track it just looks like some typos threw you off.

You first need to note that , because that would give a zero denominator.
Since the denominator is always nonnegative, you can multiply both sides by the denominator without changing the inequality sign
Quadratic inequalities are nearly always easiest solved by completing the square.
Now, remembering that , that means our solution is

Alternatively,
the fraction is undefined for
For all other x, the denominator is positive.
Then for
the solution reduces to
which will be true if both factors are negative or both are positive, or either is zero.
Both are negative if and at the same time, which is not possible.
Both are positive if
Therefore, both factors cannot be negative and so for