Answer:
Part A) The number line in the attached figure
Part B) The absolute value equation is [tex]\left|x-7\right|=3[/tex]
Step-by-step explanation:
Part A) Represent these two distances on a number line
we know that
The minimum distance between two fence posts is 4 feet, and the maximum distance is 10 feet
Let
x ----> the distance between the two fence post
so
[tex]4\leq x \leq 10[/tex]
The interval is -------> [4,10]
using a graphing tool
see the attached figure
In a number line the solution is the shaded area at right of x=4 (close circle) and at the left of x=10 (close circle)
Part B) Write an absolute value equation that represents the minimum and maximum distances
Find the midpoint of the interval [4,10]
[tex]M=(\frac{x1+x2}{2})[/tex]
substitute the values
[tex]M=(\frac{4+10}{2})[/tex]
[tex]M=(7)[/tex]
The distance from the midpoint to the endpoints of the interval is 3 feet
so
The absolute value equation is
[tex]\left|x-7\right|=3[/tex]
Verify
Solve the absolute value
case 1) positive value
[tex]+(x-7)=3[/tex]
Solve for x
[tex]x=7+3=10\ ft[/tex] ----> maximum distance
case 2) negative value
[tex]-(x-7)=3[/tex]
Solve for x
[tex]-x+7=3[/tex]
[tex]x=7-3=4\ ft[/tex] ----> minimum distance
