Solve the inequality, then graph the solution set. \[ |x-2| \geq 4 \]

To solve the inequality \( |x - 2| \geq 4 \) and graph the solution set, follow these steps: ### Step 1: Understand the Absolute Value Inequality The inequality \( |x - 2| \geq 4 \) represents all real numbers \( x \) whose distance from 2 is at least 4 units on the number line. ### Step 2: Split the Inequality An absolute value inequality of the form \( |A| \geq B \) (where \( B \geq 0 \)) can be rewritten as two separate inequalities: 1. \( A \geq B \) 2. \( A \leq -B \) Applying this to \( |x - 2| \geq 4 \): 1. \( x - 2 \geq 4 \) 2. \( x - 2 \leq -4 \) ### Step 3: Solve Each Inequality Separately 1. **First Inequality:** \[ x - 2 \geq 4 \\ x \geq 4 + 2 \\ x \geq 6 \] 2. **Second Inequality:** \[ x - 2 \leq -4 \\ x \leq -4 + 2 \\ x \leq -2 \] ### Step 4: Combine the Solutions The solutions to the inequality \( |x - 2| \geq 4 \) are all real numbers \( x \) such that: \[ x \leq -2 \quad \text{or} \quad x \geq 6 \] In **interval notation**, this is written as: \[ (-\infty, \, -2] \cup [6, \, \infty) \] ### Step 5: Graph the Solution Set Below is a graph representing the solution set on a number line: ``` <----|=================•-------------------•=================|----> -∞ -2 6 ∞ ``` - **Closed Dots at -2 and 6:** Indicate that -2 and 6 are included in the solution (≥). - **Shaded Arrows:** - To the **left of -2**: Represents all \( x \) values less than or equal to -2. - To the **right of 6**: Represents all \( x \) values greater than or equal to 6. ### Visual Representation For a clearer visual, here's a number line illustration:  *(Note: Replace the image link with an actual graph if you're presenting this in a medium that supports images.)* ### Summary - **Solution:** \( x \leq -2 \) or \( x \geq 6 \) - **Interval Notation:** \( (-\infty, \, -2] \cup [6, \, \infty) \) - **Graph:** Number line with shaded regions extending to the left of -2 and to the right of 6, including the points -2 and 6.