Question

A county government says that a safe level of chlorine in a hot tub is within 1.75 ppm of 3.25 ppm.

a. Write and solve an absolute value inequality to represent this situation.
b. A lifeguard measures the chlorine level in the pool and finds it is 1.0 ppm. Should he add more chlorine? Explain.

Answer 1

Part A : |x-2.5| ≤ 0.75 , x ∈ [1.75,3.5]

Part B : yes, the lifeguard should add more chlorine.

Explanation:

Part A:

Let C is the variation of the level of chlorine in a hot tub.

Level of chlorine in a hot tub is within 1.75 ppm of 3.25 ppm.

To find absolute value inequality, need to find the standard level of chlorine 1.75 + C or 3.25 - C

1.75 + C = 3.25 - C

2C = 5

C = 2.5

So, the standard level would be 2.5 ppm,

If x represents the present level of chlorine,

Then it would be lie within 1.75 ppm of 3.25 ppm.

1.75 ≤ x ≤ 3.25

Subtract 2.5 from all sides

1.75 - 2.5 ≤ x -2.5 ≤ 3.25 - 2.5

-0.75 ≤ (x-2.5) ≤ 0.75

which is equivalent to the following absolute value inequality.

|x-2.5| ≤ 0.75

And the solve of the inequality : x ∈ [1.75,3.5]

Part B: If x = 1.0 ppm,

∴ |1.0-2.5| = 1.5 which is not less than equal to 0.75.

Another explanation:

the minimum safe level of chlorine in a hot tub is 1.75 ppm

Since 1 < 1.75

Therefore, lifeguard should add more chlorine.