Deon is inspecting a zipline at an adventure park. For safety reasons, the top platform must be 6.5 (a) Let \( x \) represent the height (in yards) of the top platform. Write an absolute value inequality platform. These heights are where the zipline will pass inspection. - \( \square \) \( \square \) \( \square \) \( \square \) - \( \square \) \( \leq \) \( \square \) \( \square \) \( + \) \( \square \) \( > \) \( \square \) \( \square \) - \( \square \) \( > \) \( \square \) (b) Solve the absolute value inequality from part (a). \( \square \) \( \square \) and \( \square \) \( \geq \square \)

Sure, let's work through the problem step by step. ### **Problem Statement:** Deon is inspecting a zipline at an adventure park. For safety reasons, the height of the top platform must be **6.5 yards** (assuming the unit is yards based on the context). **(a)** Let \( x \) represent the height (in yards) of the top platform. Write an absolute value inequality to represent the allowable heights for the platform to pass inspection. **(b)** Solve the absolute value inequality from part (a). --- ### **Solution:** #### **Part (a): Writing the Absolute Value Inequality** To ensure the platform's height is safe, we assume there's an acceptable range around the central value of 6.5 yards. Typically, this involves a tolerance that the height can deviate from the central value. Although the exact tolerance isn't provided, for the sake of this problem, let's assume the platform height must be within **0.5 yards** of 6.5 yards. The absolute value inequality can be written as: \[ |x - 6.5| \leq 0.5 \] **Explanation:** - \( x \) is the actual height of the platform. - \( 6.5 \) is the required central height. - \( 0.5 \) is the maximum allowable deviation from 6.5 yards. #### **Part (b): Solving the Absolute Value Inequality** To solve the inequality \( |x - 6.5| \leq 0.5 \), we can break it down into a compound inequality: \[ -0.5 \leq x - 6.5 \leq 0.5 \] Now, solve for \( x \) by adding 6.5 to all parts of the inequality: \[ -0.5 + 6.5 \leq x \leq 0.5 + 6.5 \] \[ 6 \leq x \leq 7 \] **Explanation:** - The platform height \( x \) must be **at least 6 yards** and **at most 7 yards** to pass inspection. --- ### **Final Answer:** **(a)** The absolute value inequality is: \[ |x - 6.5| \leq 0.5 \] **(b)** Solving the inequality gives: \[ 6 \leq x \leq 7 \] So, the platform height must be **between 6 and 7 yards** to pass inspection.