Question

a nursery sells young palm trees. they guarantee that a palm tree will grow to be 6 feet tall, plus or minus 1.5 feet. if the trees height doesnt 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 acceptable heights of a palm tree. these are the heights where the guarantee is met.

options:

• |□ - □| > □
• |□ + □| ≤ □
• |□ - □| ≤ □
• |□ + □| > □

Explanation:

Step1: Understand the problem

The guaranteed height is 6 feet, with a tolerance of 1.5 feet. So the difference between the tree's height \( x \) and 6 feet should be within 1.5 feet (inclusive) for the guarantee to be met.

Step2: Recall absolute value inequality for tolerance

The absolute value inequality for a value \( x \) being within \( \pm k \) of a target \( a \) is \( |x - a| \leq k \). Here, \( a = 6 \) and \( k = 1.5 \), so the inequality is \( |x - 6| \leq 1.5 \). Looking at the options, the third option (with \( | - | \leq \)) matches this form.

Answer:

The third option (the one with \( |x - 6| \leq 1.5 \) - represented as \( \boxed{|x - 6| \leq 1.5} \) in the option with the \( | - | \leq \) structure)