The lawful speeds S on an interstate highway satisfy |S-54.5|<=18. Find the maximum and minimum speed limits by solving the absolute value equation |S-54.5|<=18.
To solve the absolute value inequality |S-54.5|<=18, we consider the distance between S-54.5 and 0.
Since |S-54.5| represents the distance between S-54.5 and 0, we can say that -18 <= S-54.5 <= 18.
To find the maximum speed limit, we add 54.5 to both sides of the inequality: -18 + 54.5 <= S-54.5 + 54.5 <= 18 + 54.5.
This simplifies to 36.5 <= S <= 72.
Therefore, the maximum speed limit is 72.
To find the minimum speed limit, we subtract 54.5 from both sides of the inequality: -18 - 54.5 <= S-54.5 - 54.5 <= 18 - 54.5.
This simplifies to -72.5 <= S <= -36.5.
However, speed limits are positive values, so the minimum speed limit is 0.
Therefore, the maximum speed limit is 72 mph and the minimum speed limit is 0 mph.
To find the maximum and minimum speed limits that satisfy the absolute value equation |S-54.5|≤18, we will consider the two cases:
1. S-54.5≥0 (when S-54.5 is non-negative):
In this case, the absolute value equation |S-54.5| reduces to S-54.5≤18. Adding 54.5 to both sides of the inequality, we get S≤(18+54.5), which simplifies to S≤72.5. Therefore, the maximum speed limit is 72.5 mph.
2. S-54.5<0 (when S-54.5 is negative):
In this case, the absolute value equation |S-54.5| becomes -(S-54.5)≤18. Simplifying, we get -S+54.5≤18. Subtracting 54.5 from both sides, we have -S≤18-54.5, which simplifies to -S≤-36.5. Dividing both sides of the inequality by -1 (which reverses the inequality sign), we get S≥36.5. Here, the minimum speed limit is 36.5 mph.
Therefore, the maximum speed limit is 72.5 mph, while the minimum speed limit is 36.5 mph.
To solve the absolute value inequality |S - 54.5| ≤ 18, we can break it down into two separate inequalities:
1. S - 54.5 ≤ 18
2. -(S - 54.5) ≤ 18
Solving the first inequality:
S - 54.5 ≤ 18
Add 54.5 to both sides:
S ≤ 18 + 54.5
S ≤ 72.5
Solving the second inequality:
-(S - 54.5) ≤ 18
Distribute the negative sign:
- S + 54.5 ≤ 18
Subtract 54.5 from both sides:
- S ≤ 18 - 54.5
- S ≤ -36.5
Multiply both sides by -1 (this reverses the inequality):
S ≥ -36.5
Therefore, the maximum speed limit is 72.5 mph and the minimum speed limit is -36.5 mph. However, since speed limits cannot be negative, the minimum speed limit in this case would be 0 mph.