Question

Mira's walkie-talkie has a range of 1 mile. Mira is traveling on a straight highway and is at mile marker 102. Identify the absolute-value equation and solution to find the minimum and maximum mile marker from mile marker 102 that Mira's walkie-talkie will reach.

Answer 1

The absolute-value equation to find the range of Mira's walkie-talkie from mile marker 102 is |x - 102| = 1, where x represents the mile markers. Solving this equation gives us the minimum mile marker 101 and the maximum mile marker 103 that Mira's walkie-talkie can reach.

Step-by-step explanation:

To determine the minimum and maximum mile markers that Mira's walkie-talkie will reach from mile marker 102, we can use an absolute-value equation. The range of the walkie-talkie is 1 mile, so if Mira is currently at mile marker 102, the walkie-talkie range in terms of mile markers can be written as |x - 102| = 1, where x represents the mile markers.

To find the solutions, we consider two cases based on the definition of absolute value:

Case 1: x - 102 = 1, which gives us x = 103.

Case 2: -(x - 102) = 1, which gives us x - 102 = -1, so x = 101.

Therefore, the minimum mile marker Mira can reach is 101, and the maximum is 103.

Answer 2

Assume the straight highway is along the x-axis.
Then the accessible locations of the walkie-talkie are defined by a circle of radius 1, centred on (102,0), as follows:
(x-x0)^2+(y-y0)^2=r^2
(x0,y0) is the centre of the circle = (102,0)
r=radius of range of equipment=1
so equation is
(x-102)^2+y^2<=1
in absolute-value form:
|x-102|<=|sqrt(1-y^2)|
where y is distance of any point from the highway.