Certainly! Let's solve this problem step-by-step using the concept of absolute value inequalities.
1. Understand the Problem:
You know that your car averages 31 miles per hour (mph) on the highway, but this speed can vary by at most 5 mph. You need to find the range of speeds (mileage) your car could actually be traveling.
2. Set up the Absolute Value Inequality:
The problem states that the actual speed [tex]\( v \)[/tex] can deviate from the average speed (31 mph) by at most 5 mph. This can be expressed as:
[tex]\[
|v - 31| \leq 5
\][/tex]
3. Solve the Absolute Value Inequality:
To solve [tex]\( |v - 31| \leq 5 \)[/tex], we need to convert the absolute value inequality into two separate inequalities:
[tex]\[
-5 \leq v - 31 \leq 5
\][/tex]
4. Isolate [tex]\( v \)[/tex] in the Inequalities:
To find the range of [tex]\( v \)[/tex], solve each part of the double inequality separately.
For the left part:
[tex]\[
-5 \leq v - 31
\][/tex]
Add 31 to both sides:
[tex]\[
26 \leq v
\][/tex]
For the right part:
[tex]\[
v - 31 \leq 5
\][/tex]
Add 31 to both sides:
[tex]\[
v \leq 36
\][/tex]
5. Combine the Results:
Combining these two inequalities, we get:
[tex]\[
26 \leq v \leq 36
\][/tex]
6. Conclusion:
The range of your car’s mileage on the highway, considering the variation, is between 26 mph and 36 mph.
This means your car will be traveling at any speed between 26 mph and 36 mph.
