Question

I really need help on this!

Answer 1

Q4
Shortest distance = 1 mile
Longest distance = 5 miles

Q5
1 ≤ d ≤ 5

Q6
4.5 miles

Explanation:

A diagram does help for these questions

Q4

If both Luke and Logan's houses are on the same straight line in the same direction then the difference between their individual distances from the field will be the shortest distance.

See the first diagram

Shortest distance = 3 - 2 = 1 mile

However, if they lived in exactly opposite directions on a straight line then the distance would be the longest and would be 2 + 3 = 5 miles

--------------------------------------------------------------------------------

Q5

The compound inequality would be 1 ≤ d ≤ 5 where d represents all possible distances from the two homes.

(We can also write this as [1, 5] which is called interval notation)

See second figure for how this would be represented on a number line

------------------------------------------------------------------------------

Q6

Since we have determined the greatest distance between the two houses as 5 miles, and Luke has already biked 0.5 mile he has 5 - 0.5 = 4.5 miles left to bike

Answer 2

Problem 4

I recommend to draw a number line. Place the baseball park at 0 on the number line.

Then place Luke's house at -3. The distance from -3 to 0 is 3 units, to represent the 3 miles.

Place Logan's house at 2 on the number line.

To go from -3 to 2, move 5 units to the right. This is the longest distance possible because each person's house is on opposite sides of the baseball park.

To get the shortest distance possible, we'll place each person's house to the right of 0. This time each person's house is on the same side.

We'll move Luke from -3 to +3. Keep Logan at 2. The distance from 2 to 3 is 1 unit. This is the shortest distance possible.

Answers:

Shortest distance = 1 mile

Longest distance = 5 miles

===============================================

Problem 5

d = distance between Luke's house and Logan's house

Refer to the result of problem 4, where we found d = 1 the smallest possible value of d. And also d = 5 the largest possible value.

In other words: d is between 1 and 5, including each endpoint.

The compound inequality to set up is
`1 \le d \le 5`

The graph involves closed filled in circles at 1 and 5 on the number line. Shade the region between these endpoints. Closed endpoints visually tell the reader "include this endpoint".

Answers:

Inequality:
`\boldsymbol{1 \le d \le 5}`

Graph: Closed filled in circles at 1 and 5; shading in between

===============================================

Problem 6

The largest distance we previously found was 5 miles.

Luke traveled 0.5 miles so far, meaning he has at most 5-0.5 = 4.5 miles left to go.

Answer:

4.5 miles