Question

a road sign shows a vehicles speed as the vehicle passes. complete parts (a) through (c). click the icon to view the road sign. a. the sign blinks for vehicles traveling within 5 miles per hour of the speed limit. write and solve an absolute - value inequality to find the minimum and maximum speeds of an oncoming vehicle that will cause the sign to blink. let x represent the speed of an oncoming vehicle in miles per hour. write an absolute - value inequality to find the minimum and maximum speeds of an oncoming vehicle that will cause the sign to blink.

Explanation:

Step1: Define the speed - limit relationship

Let the speed limit be $L$. The sign blinks for vehicles traveling within 5 miles per hour of the speed limit. The difference between the vehicle's speed $x$ and the speed limit $L$ should be less than or equal to 5. So the absolute - value inequality is $|x - L|\leq5$.

Step2: Solve the absolute - value inequality

Case 1: When $x - L\geq0$, the inequality $|x - L|\leq5$ becomes $x - L\leq5$. Solving for $x$ gives $x\leq L + 5$.
Case 2: When $x - L\lt0$, the inequality $|x - L|\leq5$ becomes $-(x - L)\leq5$. Expanding, we get $-x+L\leq5$. Then, $-x\leq5 - L$, and multiplying both sides by - 1 (and reversing the inequality sign) gives $x\geq L - 5$.

Answer:

The absolute - value inequality is $|x - L|\leq5$, and the minimum speed is $L - 5$ miles per hour and the maximum speed is $L + 5$ miles per hour.