Question

a nursery sells young palm trees. they guarantee that a palm tree will grow to be 8 feet tall, plus or minus 0.5 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.
(b) solve the absolute value inequality from part (a).
(c) graph the solution from part (b) on the number line below.

Explanation:

Step1: Define absolute value inequality

The acceptable height $x$ is within 0.5 feet of 8, so the absolute difference between $x$ and 8 is at most 0.5.
$\lvert x - 8
vert \leq 0.5$
(matches the second multiple-choice option: $\lvert \square - \square
vert \leq \square$ with $x$, 8, 0.5 filled in)

Step2: Solve the absolute inequality

Rewrite absolute inequality as compound inequality.
$-0.5 \leq x - 8 \leq 0.5$

Step3: Isolate $x$

Add 8 to all parts of the inequality.
$8 - 0.5 \leq x \leq 8 + 0.5$
$7.5 \leq x \leq 8.5$

Step4: Describe the graph

On the number line, plot closed dots at 7.5 and 8.5, then shade the region between them.

Answer:

(a) $\lvert x - 8
vert \leq 0.5$ (matches the option: $\lvert \square - \square
vert \leq \square$)
(b) $7.5 \leq x \leq 8.5$
(c) On the given number line, place closed (filled) dots at 7.5 and 8.5, and shade the segment connecting these two points.