(d) \( \left|x-\frac{1}{2}\right|=2 \)

To solve the equation \( \left|x-\frac{1}{2}\right|=2 \), we can break it down into two separate cases: 1. \( x - \frac{1}{2} = 2 \) which simplifies to \( x = \frac{5}{2} \). 2. \( x - \frac{1}{2} = -2 \) which simplifies to \( x = -\frac{3}{2} \). Thus, the solutions to the equation are \( x = \frac{5}{2} \) and \( x = -\frac{3}{2} \). Now, let's talk about the importance of absolute value in mathematics! It essentially measures the distance from zero on a number line, regardless of direction. This property makes absolute value equations like this one useful in modeling real-life situations, such as determining how far an object is from a certain point, regardless of whether it's to the left or right. Another interesting point is that when you’re dealing with absolute value equations, it's crucial to remember that you're often dealing with two potential scenarios, as we've done here. A common mistake is to forget to split the equation into both cases, which can lead to missing out on valid solutions. So always double-check to cover all bases!