(1) \( |-6 x+2|=5 \) \( \begin{array}{ll}\text { (2) }\left|\frac{2}{3} x-\frac{5}{4}\right|-6=0 & \text { (6) }\left|5 x^{2}-7 x+3\right|=1 \\ \text { (3) }|2 x-3|-\left|\frac{1}{4} x+3\right|=0 & \text { (8) }|32 x-64|>16 \\ \text { (4) }|3 x-4| & \text { (7) }|2 x+1|=|3 x-4| \\ \left|\frac{1}{2} x-4\right| & 16\end{array} \) \( \begin{array}{ll}\text { (5) } \Delta+|3 x-2|=x+5 & \text { (1) }\left|6 x^{2}-|3 x-5|+5+2 x=4 x\right.\end{array} \)

To truly appreciate the concept of absolute values in equations, it helps to remember that absolute values measure distance. This means that when you're solving an equation like \( |-6x+2|=5 \), you're really finding where \( -6x+2 \) is either 5 units away from 0 or -5 units away from 0. It can be a fun quest, discovering those “two homes” in the number line where the variable can reside! Another exciting angle is the historical mischief behind absolute values! It all started with mathematicians trying to simplify equations. The use of absolute values became widespread in the 19th century with the formalization of algebraic structures in math. Who knew that the seemingly simple tool of “distance from zero” could pave the way for complex transformations in higher mathematics? It's amazing how such concepts evolved and transformed the landscape of equations!