Originally Posted by

**s_ingram** $\displaystyle x - 3 < (x - 4) / x$

$\displaystyle x^2 - 3x < x - 4$

$\displaystyle x^2 - 4x + 4 < 0$

$\displaystyle (x - 2) (x - 2) < 0$

$\displaystyle (x - 2)^2 < 0 $

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:

$\displaystyle \begin{aligned}

x - 3 &< \frac{x - 4}{x} \\

x - 3 - \frac{x - 4}{x} &< 0 \\

\frac{x^2 - 3x}{x} - \frac{x - 4}{x} &< 0 \\

\frac{x^2 - 4x + 4}{x} &< 0 \\

\frac{(x - 2)^2}{x} &< 0

\end{aligned}$

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:

$\displaystyle \begin{aligned}

(x - 2)^2 &= 0 \\

x - 2 &= 0 \\

x &= 2

\end{aligned}$

I find (2) by setting the denominator equal to zero: x = 0.

Then I draw my sign chart like this:

Code:

und 0
----------+---------------+-------------
0 2

(und = undefined)

Now take the fraction

$\displaystyle \frac{(x - 2)^2}{x}$ and test a value less than 0. You will see that the fraction is negative, so I make a notation on my sign chart.

Code:

(-)^2
-----
(-) und 0
----------+---------------+-------------
neg 0 2

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 (-)^2 (+)^2
----- ----- -----
(-) und (+) 0 (+)
----------+---------------+-------------
neg 0 pos 2 pos

We want a range of numbers where the fraction is less than 0, or negative. So the solutions are x < 0.

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