As a homework, Alex was assigned to solve the exponential inequality: $$ (0.25)^{2x-1} \leq 16 $$ He handed in his homework, but the teacher returned it as incorrect.
(1) Alex modified the inequality so that it contained powers with the same base: $$ \begin{align} \left(\frac14\right)^{2x-1} & \leq 4^2 \cr \left(4^{-1} \right)^{2x-1} & \leq 4^2 \end{align} $$
(2) Further, he simplified it to: $$ 4^{-2x+1} \leq 4^2 $$
(3) Alex realized that the bases on both sides of the inequality were equal, so he removed them, and continued comparing only the exponents. He obtained: $$ -2x+1 \leq 2 $$ Since he understood that the exponential function $y=4^x$ is increasing, he did not change the inequality sign.
(4) Alex solved the newly obtained inequality as follows: $$ \begin{align} -2x+1 & \leq 2 \cr -2x & \leq 1 \cr x & \leq -\frac12 \end{align} $$
The set of all solutions is the interval $(-\infty ;-\frac12 ]$.
(5) The check is not necessary, so Alex skipped the check.
In what step did Alex make a mistake?
He made the mistake in step (2). The simplified inequality should be: $$4^{2x-2}\leq 4^2 $$
He made the mistake in step (3). He should have reversed the inequality sign and obtained: $$-2x+1 \geq 2$$
He made the mistake in step (4). The correct solution should be: $$ [ - \frac12; \infty)$$
He made the mistake in step (5). The check is an essential part of the solution.
All steps are correct except for step (4). Alex solved the inequality $$ -2x \leq 1 $$ by dividing both sides of the inequality by the negative number $(-2)$ and forgot to reverse the inequality sign. He should get: $$ x\geq-\frac12 $$ The set of all solutions of the inequality is then the interval $ [ -\frac12 ; \infty)$. The check is not necessary.