A plane is flying at a speed of 150 miles per hour. The pilot plans on flying at this speed for the next 160 miles, plus or minus 25 miles. Write an absolute value equation to find the minimum and maximum number of hours the plane will travel at that speed.

To find the minimum and maximum number of hours the plane will travel at a speed of 150 miles per hour, we can model the situation using an absolute value equation.

### Step 1: Define variables
Let \( t \) represent the time in hours that the plane will fly at 150 miles per hour.

### Step 2: Write the relationship between distance, speed, and time
The relationship is given by:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
The pilot will fly 160 miles, with a variation of ±25 miles. This means the distance can range between 135 miles and 185 miles.

### Step 3: Set up an absolute value equation
We want to write an equation for the time based on the deviation from the center distance of 160 miles. The absolute deviation from 160 miles is at most 25 miles. The equation for this situation is:
\[
|d - 160| = 25
\]
where \( d \) is the distance flown.

### Step 4: Express distance in terms of time
We know that distance is speed multiplied by time, so \( d = 150t \). Substituting into the absolute value equation, we get:
\[
|150t - 160| = 25
\]

### Step 5: Solve the equation
Now solve for \( t \):
1. Break the absolute value equation into two cases:
   \[
   150t - 160 = 25 \quad \text{or} \quad 150t - 160 = -25
   \]
2. Solve each case for \( t \):

   **Case 1**: \( 150t - 160 = 25 \)
   \[
   150t = 185
   \]
   \[
   t = \frac{185}{150} = 1.2333 \, \text{hours}
   \]

   **Case 2**: \( 150t - 160 = -25 \)
   \[
   150t = 135
   \]
   \[
   t = \frac{135}{150} = 0.9 \, \text{hours}
   \]

### Conclusion:
The plane will travel between **0.9 hours** and **1.2333 hours** at 150 miles per hour.

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Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How would the equation change if the speed of the plane varied?
2. What if the distance variation was different (e.g., ±30 miles)?
3. How would the time change if the plane flew at 200 miles per hour instead of 150?
4. Can we solve the problem using inequalities instead of absolute value?
5. How does this model relate to real-world flight conditions?

**Tip**: Always check the units to ensure the equation and results are consistent. In this case, the speed was in miles per hour, so time had to be in hours.

Learn how to solve an absolute value equation to find the minimum and maximum travel time for a plane flying at 150 miles per hour over a distance of 160 miles, with a deviation of ±25 miles. This problem applies algebraic concepts and the distance-speed-time formula. Suitable for students in Grades 8-10.

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