Algebra Problems of the Day (Integrated Algebra Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, August 2010

Part I: Each correct answer will receive 2 credits.


1. The school store did a study comparing the cost of a sweatshirt with the number of sweatshirts sold. The price was changed several times and the numbers of sweatshirts sold were recorded. The data are shown in the table below.

Which scatter plot represents the data?

Answer: 3) [See Image]

The x-axis is labeled 5, 10, 15, 20, and 25, which matches the costs of the sweatshirts, not the number sold. Note that there are points where x = 5, x = 10, etc. Eliminate Choices (2) and (4).

The first point in Choice (1) is ($5, 9 sold). That's no data from the table. Eliminate Choice (1).

The points in Choice (3) match up with the data: (10, 9), (25, 6), etc.

2. What is the solution of 3(2m − 1) ≤ 4m + 7?

1) m ≤ 5
2) m ≥ 5
3) m ≤ 4
4) m ≥ 4

Answer: 1) m ≤ 5

Solve the inequality using the Distributive Property and inverse operations.

3(2m − 1) ≤ 4m + 7

6m − 3 ≤ 4m + 7

2m − 3 ≤ 7

2m ≤ 10

m ≤ 5

Which is Choice (1).

3. Which set represents the intersection of sets A, B, and C shown in the diagram below?

1) {3, 4, 5, 6, 7}
2) {2}
3) {2, 3, 4, 5, 6, 7}
4) {1, 2, 3, 4, 5, 6, 7, 8, 9}

Answer: 2) {2}

The intersection of A, B and C is the centermost section which is contained within all three circles.

4. The end of a dog’s leash is attached to the top of a 5-foot-tall fence post, as shown in the diagram below. The dog is 7 feet away from the base of the fence post.

How long is the leash, to the nearest tenth of a foot?

1) 4.9
2) 8.6
3) 9.0
4) 12.0

Answer: 2) 8.6

The Leash is the longest side of a right triangle with legs 5 and 7. It cannot be 4.9 feet because that would be shorter than the other sides. It cannot be 12 feet because 5 - 7 - 12 is not a triangle -- it's a straight line.

You know two sides of a right triangle and you want to know the third side. That's a Pythagoean Theorem problem. (It doesn't involve basic trigonometric ratios.)

5² + 7 ² = c²

25 + 49 = c²

74 = c²

c = 8.6

You have a calculator, so use it. However, once you had 74, you should realize that 74 is between 64 and 81, so c was between 8 and 9. There's only one choice between 8 and 9.

5. What is the slope of the line passing through the points A and B, as shown on the graph below?

1) -3
2) -1/3
3) 3
4) 1/3

Answer: 2) -1/3

Rise/run. It's going down, so it's negative. It has a shallow (not steep) slope, so it's a fraction less between 0 and -1. There's only one choice.

Rise / run = -2 / 6 = -1 / 3

More to come. Comments and questions welcome.

More Regents problems.

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