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Let's go through this step by step.

### Given Information:
1. **Equation for Temperature (T) at altitude \( x \):**
   \[
   T = 84 - 19x
   \]
   where:
   - \( T \) is the temperature in °F.
   - \( x \) is the altitude in miles.

2. **Required Range of Temperature:**
   We need to find the altitudes \( x \) where the temperature \( T \) is between 4°F and 30°F.

### Part (a): Solve for \( x \) where \( 4 \leq T \leq 30 \)

Since \( T = 84 - 19x \), we can set up the inequality:
\[
4 \leq 84 - 19x \leq 30
\]

#### Step 1: Solve the left side of the inequality
\[
4 \leq 84 - 19x
\]
Subtract 84 from both sides:
\[
4 - 84 \leq -19x
\]
\[
-80 \leq -19x
\]
Divide by -19 (and remember to reverse the inequality sign because we are dividing by a negative number):
\[
\frac{-80}{-19} \geq x
\]
\[
x \leq \frac{80}{19}
\]
Calculating \( \frac{80}{19} \approx 4.21 \), so:
\[
x \leq 4.21
\]

#### Step 2: Solve the right side of the inequality
\[
84 - 19x \leq 30
\]
Subtract 84 from both sides:
\[
-19x \leq 30 - 84
\]
\[
-19x \leq -54
\]
Divide by -19 (again, reverse the inequality):
\[
x \geq \frac{-54}{-19}
\]
\[
x \geq \frac{54}{19}
\]
Calculating \( \frac{54}{19} \approx 2.84 \), so:
\[
x \geq 2.84
\]

#### Conclusion for Part (a):
The altitude \( x \) where the temperature is between 4°F and 30°F is:
\[
2.84 \leq x \leq 4.21
\]

### Part (b): Absolute Value Inequality

We can write the temperature inequality in absolute value form centered around the midpoint of the range \( [4, 30] \).

1. The midpoint of 4°F and 30°F is:
   \[
   \frac{4 + 30}{2} = 17
   \]

2. The distance from this midpoint to either endpoint (4 or 30) is:
   \[
   30 - 17 = 13
   \]

Thus, we can write:
\[
|T - 17| \leq 13
\]

Since \( T = 84 - 19x \), substitute this in:
\[
|84 - 19x - 17| \leq 13
\]
\[
|67 - 19x| \leq 13
\]

This absolute value inequality represents the range of altitudes where the temperature falls between 4°F and 30°F.

---

Would you like further details on any of these steps? 

Here are five related questions:
1. How do you solve absolute value inequalities in general?
2. How would the inequality change if the temperature range was different?
3. Can you graph the temperature equation \( T = 84 - 19x \) to visually see the altitude range?
4. What is the significance of reversing inequality signs when dividing by a negative number?
5. How would this problem change if the temperature decreased at a different rate?

**Tip:** When dealing with inequalities, remember to reverse the inequality sign whenever you multiply or divide by a negative value.

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 for altitude where the air temperature is between 4°F and 30°F, given a linear decrease in temperature. Learn how to apply linear and absolute value inequalities to solve real-world problems. Suitable for students in Grades 9-10.

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