A nursery sells young palm trees. They guarantee that a palm tree will grow to be 6.5 feet tall, plus or minus 1 foot. If the tree's height doesn't meet this guarantee after 2 years, a refund can be issued. (a) Let \( x \) represent the height (in feet) of a palm tree. Write an absolute value inequality that can be used to find the unacceptable heights of a palm tree. These are the heights where the guarantee is not met, and a refund can be issued. \( \square \) \( + \) \( \square \) \( > \) \( \square \) \( \square \) - \( \square \) \( > \) \( \square \) \( \square \) \( + \) \( \square \) \( \leq \) \( \square \) \( \square \) - \( \square \) \( \leq \) \( \square \) (b) Solve the absolute value inequality from part (a). \( \square \) \( \square \) and \( \square \) < ㅁ> ㅁ ㅁ \( \square \) \( \square \leq \square \) \( \square \geq \square \)

Sure, let's tackle both parts of the problem step by step. ### (a) Writing the Absolute Value Inequality The nursery guarantees that a palm tree will grow to be **6.5 feet tall, plus or minus 1 foot**. This means the acceptable height range for the palm trees is from **5.5 feet** to **7.5 feet**. To express the **unacceptable heights** (where the height is **not** within this range) using an absolute value inequality, we consider the distance of the tree's height from the guaranteed height (6.5 feet). If this distance is **greater than 1 foot**, the height is unacceptable. The absolute value inequality is: \[ |\, x - 6.5\,| \, > \, 1 \] **Filled-In Inequality:** \[ |\, x - 6.5\,| \, > \, 1 \] ### (b) Solving the Absolute Value Inequality To solve the inequality \( |x - 6.5| > 1 \), we break it down into two separate inequalities: 1. **First Inequality:** \[ x - 6.5 > 1 \] **Solution:** \[ x > 7.5 \] 2. **Second Inequality:** \[ x - 6.5 < -1 \] **Solution:** \[ x < 5.5 \] **Combined Solution:** \[ x < 5.5 \quad \text{or} \quad x > 7.5 \] This means that **any palm tree shorter than 5.5 feet or taller than 7.5 feet** does **not** meet the nursery's guarantee, and a **refund can be issued**. --- **Summary:** - **(a)** The absolute value inequality is \( |\, x - 6.5\,| \, > \, 1 \). - **(b)** The solution to the inequality is: \[ x < 5.5 \quad \text{or} \quad x > 7.5 \]