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To solve this problem, let’s go through each part step-by-step.

### Given Information:
The air temperature \( T \) at \( x \) miles above ground is given by:
\[
T = 84 - 19x
\]
where \( T \) is the temperature in °F, and \( x \) is the altitude in miles.

### Part (a) Determine the Altitudes \( x \) where the Temperature \( T \) is Between 4°F and 30°F

We need to find the values of \( x \) for which:
\[
4 \leq 84 - 19x \leq 30
\]

#### Step 1: Solve for \( x \) in each inequality separately.

1. **First inequality**:
   \[
   4 \leq 84 - 19x
   \]
   Subtract 84 from both sides:
   \[
   4 - 84 \leq -19x
   \]
   \[
   -80 \leq -19x
   \]
   Divide by -19 (reversing the inequality):
   \[
   \frac{80}{19} \geq x
   \]
   So, \( x \leq \frac{80}{19} \).

2. **Second inequality**:
   \[
   84 - 19x \leq 30
   \]
   Subtract 84 from both sides:
   \[
   -19x \leq 30 - 84
   \]
   \[
   -19x \leq -54
   \]
   Divide by -19 (reversing the inequality):
   \[
   x \geq \frac{54}{19}
   \]

#### Conclusion for Part (a):
The values of \( x \) where the temperature is between 4°F and 30°F are:
\[
\frac{54}{19} \leq x \leq \frac{80}{19}
\]

### Part (b) Use an Absolute Value Inequality to Describe these Altitudes

The range of \( x \) found is centered around the midpoint of \( \frac{54}{19} \) and \( \frac{80}{19} \).

#### Step 1: Find the midpoint of \( \frac{54}{19} \) and \( \frac{80}{19} \).

Midpoint:
\[
\frac{\frac{54}{19} + \frac{80}{19}}{2} = \frac{134}{38} = \frac{67}{19}
\]

#### Step 2: Determine the distance from the midpoint to each endpoint.

The distance from \( \frac{67}{19} \) to either \( \frac{54}{19} \) or \( \frac{80}{19} \) is:
\[
\left| \frac{67}{19} - \frac{54}{19} \right| = \frac{13}{19}
\]

#### Absolute Value Inequality:
The altitude \( x \) can be expressed with an absolute value inequality as:
\[
\left| x - \frac{67}{19} \right| \leq \frac{13}{19}
\]

### Final Answers:

1. **Part (a)**: \( \frac{54}{19} \leq x \leq \frac{80}{19} \)
2. **Part (b)**: \( \left| x - \frac{67}{19} \right| \leq \frac{13}{19} \)

Would you like further details or have any questions about these steps?

#### Related Questions:
1. How does changing the rate of temperature decrease affect the solution?
2. What would the range be if the temperature range changed from 4°F to 10°F instead?
3. How do we find an absolute value inequality in different contexts?
4. What happens if the ground temperature changes to a different initial value?
5. How can we apply a similar method for solving inequalities in physics problems?

#### Tip:
When dealing with absolute values, finding the midpoint and distance between the endpoints is key to forming inequalities.

Air temperature decreases as altitude increases. If the ground temperature is 84°F, then the air temperature x miles high is T = 84 – 19x.
(a) Determine the altitudes x where the temperature T is between 4°F and 30°F.
(b) Use an absolute value inequality to describe these altitudes.

Solve the altitude and temperature problem using inequalities. Given that T = 84 - 19x, determine the altitudes where temperature is between 4°F and 30°F, and express it as an absolute value inequality. Ideal for students in Grades 9-11.

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