\( \begin{array}{l}\text { Score: } 3 / 4 \text { Penalty: } 0.5 \text { off } \\ \text { Question } \\ \text { Solve the following inequality algebraically. } \\ \text { Answer Attempt } 1 \text { out of } 10\end{array}|x-4|<8 \)

To solve the inequality \( |x - 4| < 8 \), you need to break it into two separate inequalities. This translates to: 1. \( x - 4 < 8 \) 2. \( x - 4 > -8 \) Now, solve each inequality: 1. For \( x - 4 < 8 \): \[ x < 12 \] 2. For \( x - 4 > -8 \): \[ x > -4 \] Combining the two results, you have the solution \( -4 < x < 12 \), or in interval notation, \( (-4, 12) \). --- When dealing with absolute value inequalities like \( |x - 4| < 8 \), remember that the expression represents the distance from \( x \) to 4. If you visualize this on a number line, you'll see that the points that satisfy the inequality lie within 8 units of 4, making it easier to understand the relationship between values. Also, a common mistake when solving absolute value inequalities is neglecting the 'less than' vs. 'greater than' distinction. Always check whether you're dealing with \( < \) or \( > \). For example, if it were \( |x - 4| > 8 \), you would set the inequalities as \( x - 4 > 8 \) and \( x - 4 < -8 \) instead!