Solving Rational Inequalities: x < 2/(5x-1) | My Final Answer Matches Textbook!

[odolwa99]

My final answer matches that of the textbook, but do I need to change the < to > at any point as I solve this? I ask because, if I assume x is 1 (until I solve for x), then the question statement and parts of the solution are made untrue.

Homework Statement

Solve the following for x $\in\mathbb{R}:$ $\frac{x}{2x-1}<-2$

Homework Equations

The Attempt at a Solution

$\frac{x(2x-1)^2}{2x-1}<-2(2x-1)^2$
$x(2x-1)<-2(4x^2-4x+1)$
$2x^2-x<-8x^2+8x-2$
$10x^2-9x+2<0$
$(5x-2)(2x-1)=0$
$x=\frac{2}{5}$ or $\frac{1}{2}$
$\frac{2}{5}<x<\frac{1}{2}$

 

[tiny-tim]

hi odolwa99!

… if I assume x is 1 (until I solve for x), then the question statement and parts of the solution are made untrue.

but 1 isn't between 2/5 and 1/2,

so why is that a difficulty? ​

 

[odolwa99]

Yeah, I'm just double checking my work so I can be absolutely sure I'm correct at all steps. Thanks for looking at it for me.