Question

# Solve

The cruising speed of Kennedy's boat is 25 mi/h. She plans to cruise at this speed for the distances shown in the diagram. What equation models the number of hours x that Kennedy will travel ?

a. What are the minimum and maximum number of hours kennedy will travel ?

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

So, we will get two cases in the solution of the given equation. We simplify the equation as much as possible and then apply the above definition to get the value of x.

For 2nd part, we will construct an equation which models the situation with the help of the concept of absolute value.

## The correct answer is: x=2.8, 3.6

### Step by step solution:

The given equation is

|3x - 6| = 12

Using the definition of absolute value,

We get two possibilities,

For 3x - 6 < 0,

|3x - 6| = -(3x - 6) = 12

Simplifying, we get

-3x + 6 = 12

Subtracting 6 both sides, we have

-3x = 12 - 6 = 6

Dividing throughout by -3, we get

x = -2

For 3x - 6 ≥ 0,

|3x - 6| = 3x - 6 = 12

Adding 6 both sides, we get

3x = 12 + 6 = 18

Dividing by 3 throughout, we get

x = 6

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

x = -2, 6

Given,

The number of hours travelled = x

Cruising speed of Kennedy’s boat = 25 mi/h

Distance she plans to travel = 80 10 mi

We know,

Speed=

The above equation can be rewritten as

Distance=Speed × Time

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

From the above relation, we have

Total distance travelled by the boat = 25x mi

Thus, the equation representing the given situation is

|25x -80| = 10

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 25x - 80 < 0,

|25x - 80 |= -(25x - 80) = 10

Simplifying, we get

-25x + 80 = 10

Subtracting 80 both sides, we have

-25x = 10 - 80 = -70

Dividing throughout by -25, we get

x = = 2.8

For 25x - 80 ≥ 0,

|25x - 80| = 25x - 80 = 10

Adding 80 both sides, we get

25x = 10 + 80 = 90

Dividing by 25 throughout, we get

x = = 3.6

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

x=2.8, 3.6

Thus,

Maximum number of hours Kennedy travels = 3.6 hours

Minimum number of hours Kennedy travels = 2.8 hours

Using the definition of absolute value,

We get two possibilities,

Simplifying, we get

Subtracting 6 both sides, we have

Dividing throughout by -3, we get

Adding 6 both sides, we get

Dividing by 3 throughout, we get

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

Given,

The number of hours travelled = x

Cruising speed of Kennedy’s boat = 25 mi/h

Distance she plans to travel = 80 10 mi

We know,

The above equation can be rewritten as

Distance=Speed × Time

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

From the above relation, we have

Total distance travelled by the boat = 25x mi

Thus, the equation representing the given situation is

|25x -80| = 10

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 80 both sides, we have

Dividing throughout by -25, we get

Adding 80 both sides, we get

Dividing by 25 throughout, we get

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

Thus,

Maximum number of hours Kennedy travels = 3.6 hours

Minimum number of hours Kennedy travels = 2.8 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’.