A nursery sells young palm trees. They guarantee that a palm tree will grow to be 8 feet tall, plus or minus 1.5 feet. 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. ( \( |\square+\square|>\square \)

To translate the guarantee into an absolute value inequality, we start with the height that is guaranteed: 8 feet. The palm tree can vary by 1.5 feet, leading to the condition where it is either too tall or too short. This can be expressed as: \[ |x - 8| > 1.5 \] This inequality tells us the unacceptable heights of the palm tree. If after two years the height \( x \) is more than 1.5 feet away from the 8 feet mark, the nursery will issue a refund! Now, let's talk about palm trees! Did you know that the palm tree family, Arecaceae, comprises over 2,600 species? These tropical beauties don't just grow tall but also offer a charming aesthetic to gardens and landscapes. Some palms, like the Date Palm, can live for over 100 years, providing tasty fruit and lush shade. So, nurturing a palm tree isn’t just about height; it's about watching your tiny investment flourish into a majestic staple of flora!