Hi guys,
I am trying to find the values of x for which
Attached is a diagram from which it is easy to see the answer x < 0! i.e these are the values of x for which the straight line x - 3 is below the curve (which I have expressed as:
But, as we all know, inequalities can be solved in three ways: by sketching, by calculation and by completing the square. So I have tried it by calculation - here goes:
Now since (x - 2) is squared no matter what value x takes the result is always positive so it seems to me there is no solution here. Can anyone see what I am doing wrong!
I wouldn't multiply both sides by x. Instead, I would do this:
At this point I would make a sign chart. First, I need "critical points." These are values which (1) the fraction equals 0, and (2) the fraction is undefined. I find (1) by setting the numerator equal to zero:
I find (2) by setting the denominator equal to zero: x = 0.
Then I draw my sign chart like this:
(und = undefined)Code:und 0 ----------+---------------+------------- 0 2
Now take the fraction
and test a value less than 0. You will see that the fraction is negative, so I make a notation on my sign chart.
Pick a number between 0 and 2 and test the fraction, and then pick a number greater than 2 and test the fraction. When you're done, you should get this:Code:(-)^2 ----- (-) und 0 ----------+---------------+------------- neg 0 2
We want a range of numbers where the fraction is less than 0, or negative. So the solutions are x < 0.Code:(-)^2 (-)^2 (+)^2 ----- ----- ----- (-) und (+) 0 (+) ----------+---------------+------------- neg 0 pos 2 pos
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To answer both of your posts, you assumed that when you multiplied both sides by x. Remember that when you multiply both sides of an inequality by a negative number, you flip the sign. When you went from this step:
to this step:
you multiplied both sides by x, and since you didn't change the sign, you assumed that .
The problem was to solve for x, and if we pretend that we do not know what x is beforehand, we shouldn't multiply both sides by x; instead, subtract the fraction from both sides and proceed like I did earlier.
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