Inequality question

So lets see how to resolve inequalities that a bit longer than usual.

\frac{x+3}{x-1} \textless x \textless \frac{1}{2}

\frac{x+3}{x-1} \textless x and x \textless \frac{1}{2}

\frac{x+3}{x-1} -x \textless 0

\frac{x+3-x(x-1)}{x-1} \textless 0

\frac{x+3-x^2+x}{x-1} \textless 0

\frac{-x^2+2x+3}{x-1} \textless 0

\frac{(-x+3)(x+1)}{x-1} \textless 0

Number line

Number line

\Rightarrow -1 \textless x \textless 1 or x >3 and x \textless \frac{1}{2}.

Thus, -1 \textless x \textless \frac{1}{2}

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