Question

# Write and solve an absolute value equation for the minimum and maximum times for an object moving at the given speed to travel the given distance.

Hint:

### |x| is known as the absolute value of x. It is the non-negative value of x irrespective of its sign. The value of absolute value of x is given by

We will construct an equation which models the situation with the help of the concept of absolute value.

## The correct answer is: Hence, we get two values of x satisfying the given equation, x = 1.5,2.5

### Step by step solution:

Given,

Let the number of hours travelled be x

Speed of the object = 5 mi/h

Distance she plans to travel = 10 2.5 mi

We know,

The above equation can be rewritten as

We form an equation in terms of distance travelled by the boat.

From the above relation, we have

Total distance travelled by the boat = 5x mi

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 10 both sides, we have

Dividing throughout by -5, we get

Adding 10 both sides, we get

Dividing by 5 throughout, we get

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

Thus,

Maximum number of hours travelled = 2.5 hours

Minimum number of hours travelled = 1.5 hours

The above equation can be rewritten as

We form an equation in terms of distance travelled by the boat.

From the above relation, we have

Total distance travelled by the boat = 5x mi

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,

Simplifying, we get

Subtracting 10 both sides, we have

Dividing throughout by -5, we get

Adding 10 both sides, we get

Dividing by 5 throughout, we get

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

Thus,

Maximum number of hours travelled = 2.5 hours

Minimum number of hours travelled = 1.5 hours

Absolute value of a variable has many uses in mathematics. Geometrically, the absolute value of a number may be considered as its distance from zero regardless of its direction. The symbol |.| is pronounced as ‘modulus’. We read |x| as ‘modulus of x’ or ‘mod x’.