A nursery sells young palm trees. They guarantee that a palm tree will grow to be 3 feet tall, plus or minus 0.5 feet. If the tree's height doesn' 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 These are the heights where the guarantee is not met, and a refund can be issued. $$ \square $$ - $$ \square $$ $$ $$ $$ \square $$ $$ \square $$ $$ + $$ $$ \square $$ $$ \leq $$ $$ \square $$ $$ \square $$ $$ + $$ $$ \square $$ $$ $$ $$ \square $$ $$ \square $$ - $$ \square $$ $$ \leq $$ $$ \square $$ (b) Solve the absolute value inequality from part (a). $$ \square $$ and $$ \square $$ $$ \square $$ $$ \square $$ $$ \square $$ or $$ \square $$ $$ \square \leq \square $$ $$ \square $$ $$ \square $$

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A nursery sells young palm trees. They guarantee that a palm tree will grow to be 3 feet tall, plus or minus 0.5 feet. If the tree's height doesn' 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 These are the heights where the guarantee is not met, and a refund can be issued. $$ \square $$ - $$ \square $$ $$ > $$ $$ \square $$ $$ \square $$ $$ + $$ $$ \square $$ $$ \leq $$ $$ \square $$ $$ \square $$ $$ + $$ $$ \square $$ $$ > $$ $$ \square $$ $$ \square $$ - $$ \square $$ $$ \leq $$ $$ \square $$ (b) Solve the absolute value inequality from part (a). $$ \square $$ and $$ \square $$ $$ \square $$ $$ <\square $$ $$ \square $$ $$ >\square $$ $$ \square $$ or $$ \square $$ $$ \square \leq \square $$ $$ \square $$ $$ \square $$

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Let's tackle each part of the problem step-by-step. ### **Part (a): Writing the Absolute Value Inequality** The nursery guarantees that a palm tree will grow to be **3 feet tall, plus or minus 0.5 feet**. This means the acceptable height range for the palm tree is: $$ 2.5 \text{ feet} \leq x \leq 3.5 \text{ feet} $$ To express the **unacceptable heights** (where the guarantee is **not met**), we focus on heights **outside** this range. This can be represented using an absolute value inequality. The general form for an absolute value inequality that represents values **outside** a specific range is: $$ |x - \text{center}| > \text{radius} $$ Here, the **center** is **3 feet**, and the **radius** is **0.5 feet**. Plugging these values into the general form gives: $$ |x - 3| > 0.5 $$ So, the absolute value inequality is: $$ |\,x - 3\,| \, > \, 0.5 $$ ### **Part (b): Solving the Absolute Value Inequality** To solve the inequality $$ |x - 3| > 0.5 $$, we break it down into two separate inequalities: 1. **First Inequality:** $$ x - 3 > 0.5 $$ $$ x > 3.5 $$ 2. **Second Inequality:** $$ x - 3 < -0.5 $$ $$ x < 2.5 $$ Combining these two results, the solution to the inequality is: $$ x < 2.5 \quad \text{or} \quad x > 3.5 $$ **Summary of Solutions:** - **Part (a):** $$ |\,x - 3\,| \, > \, 0.5 $$ - **Part (b):** $$ x < 2.5 $$ or $$ x > 3.5 $$ **Part (a):** The unacceptable heights are represented by the absolute value inequality: $$ |x - 3| > 0.5 $$ **Part (b):** Solving the inequality: $$ x < 2.5 \quad \text{or} \quad x > 3.5 $$

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