A road sign shows a Vehicle's speed as the vehicle passes.

a. The sign blinks for vehicles travelling within  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.
b. Another sign blinks when it detects a vehicle travelling within  of a   speed limit. Write and solve an absolute value inequality to represent the speeds of the vehicles that cause the sign to blink.

The correct answer is: Hence, we get two values of x satisfying the given equation, x = 33,37

Step by step solution:
(a) Given,
Let the speed of the oncoming vehicle be x mi/h
Speed limit of the vehicles = 30 mi/h
Error allowed in the speed limit =   5 mi/h
We form an equation in terms of speed of the vehicle.
The two conditions we have are

 
And

 
We can rewrite the above equations as

And

We combine the above two cases to get the following equation.
Thus, the equation representing the given situation is

We solve the above equation to get two values of x, which will be the minimum and maximum values  of x.
Using the definition of absolute value,

We get two possibilities,

For

Simplifying, we get

 
Subtracting 30 both sides, we have

 
Dividing throughout by -1, we get

 x = 25

For  ,

Adding 30 both sides, we get

 
Thus, we get

x = 35
Hence, we get two values of x satisfying the given equation,

x = 35,25
Thus,
Maximum speed of the oncoming vehicle which makes the sign to blink = 35 mi/h
Minimum speed of the oncoming vehicle which makes the sign to blink = 25 mi/h
b) Given,
Let the speed of the oncoming vehicle be x mi/h
Speed limit of the vehicles = 35 mi/h
Error allowed in the speed limit =   2 mi/h
We form an equation in terms of speed of the vehicle.
The two conditions we have are

And

We can rewrite the above equations as

And

We combine the above two cases to get the following equation.
Thus, the equation representing the given situation is

 
We solve the above equation to get two values of x, which will be the minimum and maximum values  of x.
Using the definition of absolute value,

We get two possibilities,

For ,,

Simplifying, we get

Subtracting 35 both sides, we have

 
Dividing throughout by -1, we get

 

For  ,

Adding 30 both sides, we get

Thus, we get

x = 37
Hence, we get two values of x satisfying the given equation,

x = 33,37

|x|, which is pronounced "Mod x" or "Modulus of x," stands in for the absolute value of the variable x. The measure is the meaning of the Latin term "modulus." Common names for absolute value include numerical value and magnitude. The absolute value does not include the sign of the numeric value; it solely represents the numeric value. Any vector quantity's modulus is its absolute value and is always assumed to be positive.
Furthermore, absolute values express all quantities, including time, price, volume, and distance. Take the absolute value as an example: |+5| = |-5| = 5. The absolute value has no assigned sign. The formula to calculate a number's absolute value is |x| = x if it is greater than zero, |x| = -x if it is less than zero, and |x| = 0 if it is equal to zero.