Question

19. 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 life guard measures the chlorine level in the pool and finds it is 1.0ppm. Should he add more chlorine? Explain.

Answer 1

Answer:

a. |x - 3.25| ≤ 1.75

b. Yes, he should add more chlorine

Explanation:

1. Interpreting the statement:

Within 1.75 ppm of 3.25 ppm means that the safe level of chlorine may be in the range 3.25 ppm ± 1.75 ppm.

That is, between 3.25 ppm - 1.75 ppm and 3.25 ppm + 1.75 ppm (including the extreme values).

2. Absolute value inequality:

From that, the mid point of the safe interval is 3.25 ppm.

So, you use such mid point to build the absolute value inequality to represent the situation:

• |x - 3.25| ≤ 1.75

3. A life guard measures the chlorine level in the pool and finds it is 1.0ppm. Should he add more chlorine?

Solve the inequality, removing the bars of the absolute value function in this way:

• - 1.75 ≤ x - 3.25 ≤ 1.75

Add 3.25 to all the parts:

• -1.75 + 3.25 ≤ x - 3.25 + 3.25 ≤ 1.75 + 3.25

• 1.50 ≤ x ≤ 5.00

Then, the level of chlorine should be between 1.50 ppm and 5.00 ppm (including both extremes).

Hence, you conclude that the life guard should add more chlorine because 1.0 ppm is below 1.50 ppm, which is the lower bound of the safe level.