Solve and graph the solution. \[ |x| \leq 9 \]

To solve and graph the inequality \( |x| \leq 9 \), let's follow these steps: ### **1. Understanding the Absolute Value Inequality** The inequality \( |x| \leq 9 \) represents all real numbers \( x \) whose distance from 0 on the number line is **less than or equal to** 9. ### **2. Solving the Inequality** The absolute value inequality \( |x| \leq 9 \) can be rewritten without the absolute value as a compound inequality: \[ -9 \leq x \leq 9 \] **Explanation:** - \( |x| \leq 9 \) means that \( x \) is at most 9 units away from 0 in either direction on the number line. - Therefore, \( x \) can range from -9 to 9, inclusive. ### **3. Graphing the Solution** To graph the solution \( -9 \leq x \leq 9 \) on a number line: 1. **Draw a horizontal line** to represent the number line. 2. **Mark the points** at -9 and 9 on the number line. 3. **Shade the region** between -9 and 9 to indicate all the possible values of \( x \) that satisfy the inequality. 4. **Use closed circles** (●) at -9 and 9 to show that these endpoints are included in the solution (since the inequality is "less than or *equal to*"). Here's a visual representation: ``` <-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----> -12 -9 -6 -3 0 3 6 9 12 ●-----------------------------● ``` - The **solid circles** at -9 and 9 indicate that these values are included. - The **shaded line** between -9 and 9 represents all real numbers \( x \) such that \( -9 \leq x \leq 9 \). ### **4. Interval Notation** The solution can also be expressed in interval notation as: \[ [-9, \, 9] \] ### **Summary** - **Solution:** All real numbers \( x \) such that \( -9 \leq x \leq 9 \). - **Graph:** A number line with a shaded segment from -9 to 9, including the endpoints. --- **Visual Graph:** ``` Number Line: <-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----> -12 -9 -6 -3 0 3 6 9 12 Shaded Interval: ●=============================● -9 9 ``` - **●** represents a closed circle (endpoint included). - **=============================** represents the shaded region indicating all values between -9 and 9 are solutions.