1. **State the problem:** Solve the inequality $|2x+1| \leq 0$. 2. **Recall the property of absolute value:** The absolute value $|A|$ is always greater than or equal to zero for any expression $A$. It equals zero if and only if $A=0$. 3. **Apply this to the inequality:** Since $|2x+1| \leq 0$ and absolute values cannot be negative, the only way this inequality holds is if $|2x+1|=0$. 4. **Set the inside of the absolute value to zero:** $$2x+1=0$$ 5. **Solve for $x$:** $$2x=-1$$ $$x=\frac{-1}{2}$$ 6. **Conclusion:** The solution to $|2x+1| \leq 0$ is $x=-\frac{1}{2}$. This means the expression inside the absolute value must be exactly zero for the inequality to hold.