# How do I solve this inequality?

• Dec 22nd 2010, 05:52 PM
dd86
How do I solve this inequality?
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?

• Dec 22nd 2010, 05:56 PM
dwsmith
$\mbox{multiply (x-2) and then add 3} \ \ |x|<2x(x-2)+3\Rightarrow |x|<2x^2-4x+3$

$\Rightarrow x<2x^2-4x+3 \ \mbox{and} \ -x<2x^2-4x+3\Rightarrow -2x^2+4x-3

$|x| \ \mbox{is} \ -x \ \forall x\in\mathbb{R}^- \ \mbox{and} \ x \ \forall x\in\mathbb{R}^+$
• Dec 22nd 2010, 06:55 PM
Prove It
You need to consider a few cases, since when you multiply by a negative number, the inequality sign changes direction.

$\displaystyle \frac{|x| - 3}{x - 2} < 2x$.

First, it should be clear that $\displaystyle x - 2 \neq 0 \implies x \neq 2$.

Case 1: $\displaystyle x - 2 > 0 \implies x > 2$, then

$\displaystyle |x| - 3 < 2x(x-2)$

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

Since $\displaystyle x > 2$ and is therefore positive, we can say $\displaystyle |x| = x$, so

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

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

$\displaystyle 2x^2 - 5x + 3 > 0$

$\displaystyle x^2 - \frac{5}{2}x + \frac{3}{2} > 0$

$\displaystyle x^2 - \frac{5}{2}x + \left(-\frac{5}{4}\right)^2 - \left(-\frac{5}{4}\right)^2 + \frac{3}{2} > 0$

$\displaystyle \left(x - \frac{5}{4}\right)^2 - \frac{25}{16} + \frac{24}{16} > 0$

$\displaystyle \left(x - \frac{5}{4}\right)^2 - \frac{1}{16} > 0$

$\displaystyle \left(x - \frac{5}{4}\right)^2 > \frac{1}{16}$

$\displaystyle \left|x - \frac{5}{4}\right| > \frac{1}{4}$

$\displaystyle x - \frac{5}{4} < -\frac{1}{4}$ or $\displaystyle x - \frac{5}{4} > \frac{1}{4}$

$\displaystyle x < 1$ or $\displaystyle x > \frac{3}{2}$.

Since we know that $\displaystyle x > 2$, we can say $\displaystyle x > 2$ satisfies the original inequality.

Case 2: $\displaystyle x - 2 < 0 \implies x < 2$, then

$\displaystyle |x| - 3 > 2x(x - 2)$

$\displaystyle |x| - 3 > 2x^2 - 4x$.

Since $\displaystyle x < 2$, we need to consider that $\displaystyle |x| = x$ if $\displaystyle 0 \leq x < 2$ and $\displaystyle |x| = -x$ if $\displaystyle x < 0$.

Case 2a) $\displaystyle 0 \leq x < 2 \implies |x| = x$, then

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

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

$\displaystyle 2x^2 - 5x + 3 < 0$

$\displaystyle x^2 - \frac{5}{2}x + \frac{3}{2} < 0$

$\displaystyle \left(x - \frac{5}{4}\right)^2 - \frac{1}{16} < 0$

$\displaystyle \left(x - \frac{5}{4}\right)^2 < \frac{1}{16}$

$\displaystyle \left|x - \frac{5}{4}\right| < \frac{1}{4}$

$\displaystyle -\frac{1}{4} < x - \frac{5}{4} < \frac{1}{4}$

$\displaystyle 1 < x < \frac{3}{2}$.

Since we know $\displaystyle 0 \leq x <2$, that means $\displaystyle 1 < x < \frac{3}{2}$ satisfies the inequality.

Case 2b) $\displaystyle x < 0 \implies |x| = -x$, then

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

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

$\displaystyle 2x^2 - 3x + 3 < 0$

$\displaystyle x^2 - \frac{3}{2}x + \frac{3}{2} < 0$

$\displaystyle x^2 - \frac{3}{2}x + \left(-\frac{3}{4}\right)^2 - \left(-\frac{3}{4}\right)^2 + \frac{3}{2} < 0$

$\displaystyle \left(x - \frac{3}{4}\right)^2 - \frac{9}{16} + \frac{24}{16} < 0$

$\displaystyle \left(x - \frac{3}{4}\right)^2 + \frac{15}{16} < 0$.

Since the LHS is always positive, this inequality is never true.

So putting it all together, the solution to the inequality

$\displaystyle \frac{|x| - 3}{x - 2} < 2x$ is

$\displaystyle x\in \left(1, \frac{3}{2}\right) \cup \left(2, \infty\right)$.
• Dec 23rd 2010, 03:53 AM
dd86
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?
• Dec 23rd 2010, 04:03 AM
dwsmith,
can you multiply both sides by $(x-2)$ without knowing whether it is positive or not?
• Dec 23rd 2010, 04:43 AM
dd86
Quote:

dwsmith,
can you multiply both sides by $(x-2)$ without knowing whether it is positive or not?

Could this be the reason why we have to multiply the square of the denominator on both sides to remove it?
• Dec 23rd 2010, 12:10 PM
Alternatively...

$\displaystyle\frac{|x|}{x-2}-\frac{3}{x-2}<2x\Rightarrow\ \frac{|x|}{x-2}<\frac{3}{x-2}+2x\Rightarrow\frac{|x|}{x-2}<\frac{2x^2-4x+3}{x-2},\;\;\;x\ \ne\ 2$

The 3 cases are

$(1)\;\;x>2,\;\;\;(2)\;\;0

(1)

$\displaystyle\ x-2>0\Rightarrow\frac{x}{x-2}<\frac{2x^2-4x+3}{x-2}\Rightarrow\ 2x^2-5x+3>0$

$\Rightarrow\ (2x-3)(x-1)>0,\;\;x>2$

Both factors must be positive or both must be negative while $x>2$

$\displaystyle\Rightarrow\ x>\frac{3}{2},\;\;x>1,\;\;x>2\Rightarrow\ x>2$

Or

$\displaystyle\ x<\frac{3}{2},\;\;x<1,\;\;x>2$ .... impossible

(2)

$x-2<0,\;\;\;|x|=x$

We will be multiplying both sides by a negative value, so we reverse the inequality sign.

$x>2x^2-4x+3\Rightarrow\ (2x-3)(x-1)<0$

$\displaystyle\ x>\frac{3}{2},\;\;x<1$ ... impossible

$\displaystyle\ x<\frac{3}{2},\;\;x>1$ ... a valid solution.

(3)

$x<0\Rightarrow\ |x|=-x,\;\;x-2<0$

$\displaystyle\Rightarrow\frac{-x}{x-2}<\frac{2x^2-4x+3}{x-2}\Rightarrow\ 2x^2-4x+3<-x\Rightarrow\ 2x^2+3x+3<0$

For this quadratic, $f(0)=3$ and it's roots are complex, since they are $\displaystyle\frac{-3\pm\sqrt{3^2-4(2)3}}{2(2)}$

$\Rightarrow\ 2x^2+3x+3>0$

The solution is

$x>2,\;\;\;\displaystyle\ 1
• Dec 24th 2010, 06:19 AM
dd86
Thanks to all your replies, I've managed to get the answer given by my text. Your approaches widened my perspective.

If anyone would like to know how I got the answer, I'd be more than glad to do so here.