Linear inequalities in one or two variables

During a portion of a flight, a small airplane's cruising speed varied between `150` miles per hour and `170` miles per hour, inclusive. If `s` represents the cruising speed, in miles per hour, during this portion of the flight, which of the following inequalities best represents this situation?

A) `s ≥ 170`

B) `s ≤ 150`

C) `s = 160`

D) `150 ≤ s ≤ 170`

Answer: D 

"Varied between `150` miles per hour and `170` miles per hour, inclusive" means the speed `s` was greater than or equal to `150 (s ≥ 150)` AND less than or equal to `170 (s ≤ 170)`. This is best represented by the compound inequality `150 ≤ s ≤ 170`.

Ty set a goal to walk at least `24` kilometers every day to prepare for a multi-day hike. On a certain day, Ty plans to walk at an average speed of `4` kilometers per hour. What is the minimum number of hours Ty must walk on that day to fulfill the daily goal?

A) `4`

B) `6`

C) `20`

D) `24`

Answer: B

It is given that Ty plans to walk at an average speed of `4` kilometers per hour. The number of kilometers Ty will walk is determined by the expression `4s`, where `s` is the number of hours Ty walks. The given goal of at least `24` kilometers means that the inequality `4s >= 24` represents the situation. Dividing both sides of this inequality by `4` gives `s >= 6`, which corresponds to a minimum of `6` hours Ty must walk.

`y > 4x + 8`

For which of the following tables are all the values of `x` and their corresponding values of `y` solutions to the given inequality?

Answer: A

You can test the first pair of values from each table in the inequality. For table A, when `x=2`, the inequality becomes `y > 4(2) + 8`, which simplifies to `y > 16`. The table gives `y=19`, and since `19 > 16`, this point is a solution. Checking the other points: for `x=4`, `y=30` and `30 > 24` (True); for `x=6`,`y=41` and `41 > 32` (True). Since all points in table A work, it is the correct answer.