3. Solve the inequality:

$$
\frac{2|2 x-1|+2}{3}+\frac{6}{1+|2 x-1|} \leq 4-\sqrt{16 x^{4}-8 x^{2}+1}
$$
# Solution:

The domain of the inequality is all real numbers. Rewriting the left side of the inequality as $2\left(\frac{|2 x-1|+1}{3}+\frac{3}{1+|2 x-1|}\right)$, we notice that it is not less than 4, as it is twice the sum of two positive reciprocals, and it equals 4 only when $1+|2 x-1|=3$.

At the same time, the right side of the inequality

$4-\sqrt{16 x^{4}-8 x^{2}+1}=4-\sqrt{\left(4 x^{2}-1\right)^{2}}=4-\left|4 x^{2}-1\right| \leq 4 . \quad$ Therefore, $\quad$ the inequality is equivalent to the system of equations:

$$
\left\{\begin{array} { c } 
{ 1 + | 2 x - 1 | = 3 } \\
{ 4 - | 4 x ^ { 2 } - 1 | = 4 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ | 2 x - 1 | = 2 } \\
{ | 4 x ^ { 2 } - 1 | = 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{c}
x=-0.5 ; 1.5 \\
x= \pm 0.5
\end{array} \Leftrightarrow x=-0.5\right.\right.\right.
$$

Answer: $x=-0.5$.

| Points | Criteria for Grading |  |  |  |
| :---: | :--- | :--- | :--- | :---: |
| 15 | The correct answer is obtained with justification |  |  |  |
| 12 | The answer is incorrect due to an arithmetic error in an otherwise justified solution |  |  |  |
| 10 | Both sides of the inequality are correctly estimated and/or the problem is reduced to an equivalent system of equations |  |  |  |
| 5 | One side of the inequality is correctly estimated |  |  |  |
| 0 | The solution does not meet any of the above criteria |  |  |  |