The Johnson Family is looking to buy a new house on Belmont Road. Their insurance deductible is increased by $500 if they live more than 2 miles away from the fire department.

a. Write an absolute value inequality that represents the distance from the fire department that will be the best location on Belmont Road for the Johnson's to buy their home if they don't want to pay the increased deductible. We will call the location of the fire department 0 miles with Belmont Road extending away from the fire department in each direction.

b. Solve the inequality.

a) Absolute Value Inequality = > Absolute (0 + y) < 2

b) - 2 < y < 2

Which means, Johnson Family has to live within the range of - 2 to + 2 from the fire department. Otherwise, they will have to pay 500 USD as increased deductible.

Explanation:

Johnson Family has to live within the range of - 2 to + 2 from the fire department.

a) Absolute Value Equation:

Absolute (0 + y) < 2

where y represent the location of the new house and 0 represents the location of the fire department.

Furthermore,

Absolute (0 + y) < 2 = (0 + x) < 2 when (0 + y) is + ve.

and

Absolute (0+y) <2 = - (0 + x) < 2 when (0 + y) is - ve.

b) When (0 + y) is + ve,

we have, (0 + y) < 2.

Solving for y and subtracting 0 from both sides.

0-0 + y < 2 - 0

y < 2

and when (0 + y) is - ve,

we have, - (0 + y) < 2.

Solving for y:

- 0 - y < 2

multiplying negative from both sides

y > - 2

So, we have - 2 < y < 2

Johnson Family has to live within the range of - 2 to + 2 from the fire department. Otherwise, they will have to pay 500 USD as increased deductible.