\[ \begin{array}{l} |2 x-2|=4 \\ x=\square \\ x=\square \end{array} \] (smaller value) (larger value) Additional Materials

To solve the equation \(|2x - 2| = 4\), we can break it into two separate cases. First, let's set up the two scenarios from the absolute value expression: 1. \(2x - 2 = 4\) 2. \(2x - 2 = -4\) For the first case, add 2 to both sides to get \(2x = 6\) and then divide by 2, yielding \(x = 3\). For the second case, add 2 to both sides to get \(2x = -2\) and divide by 2, yielding \(x = -1\). So, the solutions are \(x = -1\) (smaller value) and \(x = 3\) (larger value). --- You might be surprised to learn that absolute value equations can model a variety of real-life scenarios. For example, in physics, they can represent distances, such as how far away an object is from a certain point, regardless of direction. When you're calculating speeds or determining how far you need to travel, these equations help clarify the reality of motion without worrying about whether you're moving forward or backward. --- When working with absolute values, a common mistake is forgetting to consider both positive and negative scenarios of the equation. Always remember to set the expression inside the absolute value equal to both the positive and negative values of the other side to find all solutions! It’s a simple step that can lead you to miss half the answers if overlooked.