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}&lt;-2

Homework Equations

The Attempt at a Solution

\frac{x(2x-1)^2}{2x-1}&lt;-2(2x-1)^2
x(2x-1)&lt;-2(4x^2-4x+1)
2x^2-x&lt;-8x^2+8x-2
10x^2-9x+2&lt;0
(5x-2)(2x-1)=0
x=\frac{2}{5} or \frac{1}{2}
\frac{2}{5}&lt;x&lt;\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.