algebraic equations

Question

Jevonte has a points card for a movie theater.

1. He receives 40 rewards points just for signing up.
2. He earns 2.5 points for each visit to the movie theater.
3. He needs at least 65 points for a free movie ticket.

Use the drop-down menu below to write an inequality representing vv, the number of visits he needs to make in order to get a free movie ticket.

Answer 1

Let's go through the problem systematically and keep track of important information in bold like this.

He receives 40 rewards points just for signing up.

The problem doesn't say whether Jevonte has signed up already. We're probably supposed to assume he just signed up today for the first time.

Points = ??? + 40

He earns 2.5 points for each visit to the movie theater.

We don't know how many visits he'll be making, so that's a variable.

If he makes one visit, he gets 2.5 points. If he makes two visits, he gets twice as many, 5. And so on.

Every time he visits, he gets another 2.5 points in addition to the 40 he got when he signed up.

Points = 40 + 2.5v

(Bonus: Notice that this looks like a line! Every visit he gets the same number of points added to his total, that's a line with a constant slope.)

He needs at least 65 points for a free movie ticket.

We can think about what happens if he has more points than that later, for now let's focus on the scenario where he has juuuust enough points.

We substitute 65 into our earlier equation for the number of points to find out how many visits he has to make to get 65 points.

65 = 40 + 2.5v

Solve for v:

65-40=2.5v

25=2.5v

25/2.5=v

10=v

To get exactly enough points, he needs to visit 10 times.

He could also get extra points and still get the reward.

He gets extra points for extra visits.

As long as v ≥ 10, he gets a reward.

(They won't take his reward away if he visits more than the minimum, that would be a badly designed incentive program.)

Optional Bonus:

If the problem gave us different numbers, we might have had v with a fractional remainder.

If he only got 39 points for signing up, then:

24/2.5=v (one less point on the left hand side)

9.6=v

It's not possible to make 0.6 visits to the theater, either you go or you don't.

How do you think we should solve problems like this?

Answer 2

1. Let's express the number of movie visits and the number of rewards points as follows:

visits = V; points = P

2. We know that 40 initial points are rewarded before any movie visits occur:

P = 40 (0 visits)

3. We know that for every visit, 2.5 points will be achieved. So 1 visit = 2.5 points, 2 visits = 5 points, 3 visits = 7.5 points, and so forth. We can equate the relationship of 2.5*V = P to be added to the additional 40:

2.5*V + 40 = P (1)

3. We know that 65 points are desired for a free ticket, thus we can set P=65 in equation #1:

2.5*V + 40 = 65

4. We know that more than 65 points can be achieved, we just need 65 points or more to achieve a free ticket:

2.5*V + 40 ≥ 65 (2)

5. Equation #2 represents the inequality representing the relationship between the number of visits needed to achieve AT LEAST 65 points. Solving for V in equation #2 can get us the inequality representing the exact quantity of visits needed:

2.5*V ≥ 65 - 40

2.5*V ≥ 25

V ≥ 10

6. Therefore, a minimum of 10 visits is required to earn a free movie ticket. Using equation #1, we can further calculate the number of visits needed to achieve further points to achieve additional benefits.

Answer 3

mx+b >= Points needed.

40 + 2.5v >= 65

You didn't provide the answer choices, so I can't tell you the correct choice.

It could be what's above, or it could be 2.5v >= 25 or v >= 10.

Answer 4

Let x be the number of movie theater visits

Fixed number of reward points + Reward points per visit * number if visits ≥ Number of points for a free ticket

40 + 2.5x ≥ 65