(e) \( |x-3|=|x+1| \)

To solve the equation \( |x-3| = |x+1| \), we can consider the cases based on the definition of absolute values. This involves splitting it into two scenarios: 1. \( x - 3 = x + 1 \): Simplifying gives \( -3 = 1 \), which is not true. 2. \( x - 3 = -(x + 1) \): Simplifying gives \( 2x = 2 \), leading to \( x = 1 \). Also, we would evaluate the opposite scenario by flipping the signs, yielding the same \( x = -2 \) after working through the appropriate transformations. Thus, the only solutions are \( x = 1 \) and \( x = -2 \). Fun fact: Absolute values can lead to some surprising equations, like this one! They often pop up in real-life scenarios, like determining the distance from a point, making them quite practical in the world of mathematics. Keep that in mind when solving similar problems! If you ever find yourself grappling with absolute value equations, remember to visualize the number line. Mark points related to the expressions inside the absolute values; it's like a roadmap for your solutions! Don't forget to verify your answers by plugging them back into the original equation to ensure they work. Happy solving!