Solve the following inequality algebraically. \[ 5|x+2|-1>24 \]

To solve the inequality algebraically: \[ 5|x + 2| - 1 > 24 \] **Step 1: Isolate the Absolute Value** Add 1 to both sides to isolate the term with the absolute value: \[ 5|x + 2| > 25 \] **Step 2: Divide Both Sides by 5** \[ |x + 2| > 5 \] **Step 3: Understand the Absolute Value Inequality** The inequality \( |x + 2| > 5 \) means that the expression inside the absolute value is either greater than 5 or less than -5. This gives us two separate inequalities: 1. \( x + 2 > 5 \) 2. \( x + 2 < -5 \) **Step 4: Solve Each Inequality** 1. **First Inequality:** \[ x + 2 > 5 \\ x > 5 - 2 \\ x > 3 \] 2. **Second Inequality:** \[ x + 2 < -5 \\ x < -5 - 2 \\ x < -7 \] **Step 5: Combine the Solutions** The solutions \( x > 3 \) and \( x < -7 \) can be combined to express all values of \( x \) that satisfy the original inequality. **Final Solution:** \[ x < -7 \quad \text{or} \quad x > 3 \] **In Interval Notation:** \[ (-\infty, -7) \cup (3, \infty) \]