Question

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. simplify your answers. solve the absolute - value inequality. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the solution is (xleq) b. the solution is (x >) or (xleq) c. the solution is (xgeq) d. the solution is (33leq xleq37) e. the solution is all real numbers. f. there is no solution. 2. the sign is programmed to blink using absolute - value inequalities of the form (|x - a|leq b) and (|x - a|geq b). which of these formulas is used to program the sign for cars traveling either 5 miles per hour above or below the 20 - miles - per - hour speed limit? what are the values of (a) and (b)? explain. the formula is used to program the sign for cars traveling either 5 miles per hour above or below the 20 - miles - per - hour speed limit. the value of (a) is and the value of (b) is . (|) represents the distance of a measurement from a fixed value, and is that fixed value.

Explanation:

Step1: Analyze absolute - value inequality concept

The general form of an absolute - value inequality for values within a certain range around a central value \(a\) is \(|x - a|\leq b\), which is equivalent to \(-b\leq x - a\leq b\) or \(a - b\leq x\leq a + b\).

Step2: Consider the speed - limit example

If the speed limit is \(20\) miles per hour and the sign blinks for speeds \(5\) miles per hour above or below the limit, then \(a = 20\) (the central speed - limit value) and \(b = 5\). The inequality \(|x - 20|\leq5\) is used. Expanding it: \(- 5\leq x - 20\leq5\), adding \(20\) to all parts gives \(15\leq x\leq25\).

Step3: Relate to the given solution \(33\leq x\leq37\)

For the solution \(33\leq x\leq37\), we can write it in the absolute - value form. The mid - point \(a=\frac{33 + 37}{2}=35\) and the distance from the mid - point to either end \(b = 2\). The absolute - value inequality is \(|x - 35|\leq2\).

Answer:

For the first part, if the solution is \(33\leq x\leq37\), the absolute - value inequality \(|x - 35|\leq2\) represents it. For the speed - limit part: The formula \(|x - a|\leq b\) is used to program the sign for cars traveling within a certain range of the speed limit. Here, if the speed limit is \(20\) and the range is \(5\) miles per hour above or below, \(a = 20\) and \(b = 5\). The absolute - value inequality \(|x - 20|\leq5\) is used, where \(|x - a|\) represents the distance of a measurement (\(x\), the speed of the vehicle) from a fixed value (\(a\), the speed limit), and \(a\) is the fixed value.