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

Ask by Clarke Mcguire. in the United States
Feb 19,2025

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Answer

The solution to the inequality \( |x-4|<8 \) is \(-4 < x < 12\).

Solution

Solve the equation \( |x-4|<8 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\left|x-4\right|<8\) - step1: Separate into possible cases: \(\left\{ \begin{array}{l}x-4<8\\x-4>-8\end{array}\right.\) - step2: Solve the inequality: \(\left\{ \begin{array}{l}x<12\\x>-4\end{array}\right.\) - step3: Find the intersection: \(-4

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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!

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