Thursday, December 02, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2011)

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 2011

16. The length of a rectangle is 3 inches more than its width. The area of the rectangle is 40 square inches. What is the length, in inches, of the rectangle?

1) 5
2) 8
3) 8.5
4) 11.5

Honestly, as a multiple-choice question, this one is boring and almost too obvious. Area is length times width, and if you factor 40, the first pairs you're likely to come up with are 10 and 4, and 5 and 8. The latter pair have a difference of 3, as stated in the question.

The length is bigger, according to the question, so the length is 8.

The width would be 5.

Had this not been multiple choice, you would've had to have set up a quadratic equation for the area and solved it ... except that the answer was "obviously" 5 and 8.

17. In interval notation, the set of all real numbers greater than −6 and less than or equal to 14 is represented by

1) (−6, 14)
2) [−6, 14)
3) (−6, 14]
4) [−6, 14]

Interval notation is a leftover from the days when mathematicians did everything by hand including keeping their notes, so this was an easy shorthand to use. It could be a little confusing in that sometimes it looks like an ordered pair.

Notice that "greater than" -6 and "less than or equal to" 14 are specified. One has equal and the other does not. That should tell you that the two symbols should be different. So eliminate Choices (1) and (4).

Now if you look at ")" and "]", which one would you think meant both "greater than or equal to"? If you think about it, you might decide that the square bracket is like that closed circle on the number line and that the curved parentheses would be more like the open circle.

If you thought that, you would be correct. The correct answer is Choice (3).

This is one of those rare occasions when what is basically a defintion question can be reasoned out.

18. Which equation represents a quadratic function?

1) y = x + 2
2) y = | x + 2 |
3) y = x2
4) y = 2x

This is another definition question. However, since so much of the material in Algebra deals with quadratic equations, it shouldn't be a difficult one.

Choice (1) is a simple linear function.

Choice (2) is an absolute value function.

Choice (3), with the exponent of 2, is a quadratic function.

CHoice (4), with the exponent of x, is an exponential function.

19. Ben has four more than twice as many CDs as Jake. If they have a total of 31 CDs, how many CDs does Jake have?

1) 9
2) 13
3) 14
4) 22

Jake has fewer CDs than Ben. This means that he couldn't have 22 and probably won't have 14 either. You can set up a system of equations for this, but it seems easier to check the choices.

If Jake has 9, then how much is 2(9) + 4? It's 22. Doess 22 + 9 = 31? Yes it does. This is the answer.

If Jake had 13, then 2(13) + 4 = 30. But there are only 31 CDs altogether, so this is already too big. And no need to go any farther.

20.What are the roots of the equation x2 − 5x + 6 = 0?

1) 1 and -6
2) 2 and 3
3) -1 and 6
4) -2 and -3

This is a good "trick" question in that the roots change dramatically depending on whether the final sign (before the 6) is a plus or a minus.

Since it is "+ 6", we know the roots will be the same sign. Since the first sign is negative, we know that the factors are negative, but the roots themselves will be positive.

If you substitute x = 1, you will see Choice (1) is incorrect. 1 - 5 + 6 != 0

If you substitute x = 2, you will see Choice (2) is the answer. 4 - 10 + 6 = 0. Since x = 2 is a solution and no other choice has x = 2 as a root, this must be the answer.

If you substitute x = 6 (skipping over the signed number for a moment), 36 - 30 + 6 != 0, so Choice (3) is incorrect.

If you substitute x = -2, 4 - (-10) + 6 != 0, so Choice (4) is incorrect.

x2 - 5x + 6 factors into (x + 2)(x + 3) = 0, so x = -2 or x = -3 must this statement true.

More to come. Comments and questions welcome.

More Regents problems.

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