Four students, Eve, Paula, Jane, and Diana, solved the inequality: $$ \left(\frac12 \right)^{x-1}<\frac12 $$ Each of them started to solve it in her own way. Which of them proceeded correctly in simplifying the inequality?
Eve: $$ \begin{gather} 2^{-x+1}>2^{-1} \cr -x+1>-1 \end{gather} $$
Diana: $$ \begin{gather} \left(\frac12 \right)^{x-1}<\left(\frac12 \right)^1 \cr x-1<1 \end{gather} $$
Paula: $$ \begin{gather} \left(\frac12 \right)^{x-1}<\left(\frac12 \right)^1 \cr x-1>1 \end{gather} $$
Jane: $$ \begin{gather} \left(\frac12 \right)^x \cdot \left(\frac12 \right)^{-1}<\left(\frac12 \right) \cr \left (\frac12 \right)^x<0 \end{gather} $$
Paula
Eve
Diana
Jane
None of them
Paula started to solve the inequality correctly.
Eve modified the base from $\frac12$ to $2$ and changed the inequality even though she should not have changed it.
Diana did not realize the change of inequality in the exponential inequality with the base between $0$ and $1$.
Jane made the mistake in converting the expression $\left(\frac12 \right)^{-1}$ to the right-hand side of the inequality. Correctly, the right-hand side should be $(\frac12 )^2$ instead of $0$.