Algebra Problems of the Day (Integrated Algebra Regents, June 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, June 2011

Part I: Each correct answer will receive 2 credits.


21. An example of an algebraic expression is

1) y = mx + b
2) 3x + 4y − 7
3) 2x + 3y ≤ 18
4) (x + y)(x − y) = 25

Answer: 2) 3x + 4y − 7

An expression has only variables, constants and operators. There are no equal signs or inequality symbols in an expression.

If you forgot this (which is really a basic definition), you might've guessed from which one doesn't belong.

22. A study showed that a decrease in the cost of carrots led to an increase in the number of carrots sold. Which statement best describes this relationship?

1) positive correlation and a causal relationship
2) negative correlation and a causal relationship
3) positive correlation and not a causal relationship
4) negative correlation and not a causal relationship

Answer: 2) negative correlation and a causal relationship

Note that this says CAUSAL (as is something causes something else) not CASUAL. Misreading this will leave you scratching your head.

The cost of carrots and the number sold have a negative correlation. If the price goes up, fewer are sold. If the price drops, more are sold. That is cause and effect.

23. Given: A = {3, 6, 9, 12, 15}
B = {2, 4, 6, 8, 10, 12}

What is the union of sets A and B?

1) {6}
2) {6, 12}
3) {2, 3, 4, 8, 9, 10, 15}
4) {2, 3, 4, 6, 8, 9, 10, 12, 15}

Answer: 4) {2, 3, 4, 6, 8, 9, 10, 12, 15}

The union is the set containing all the elements that are in both sets (without duplicates). That would be Choice (4), which has all the elements and nothing that shouldn't be there.

Choice (2) is the intersection of the sets and Choice (3) isn't the union because it's missing 6 and 12.

24. The value of a car purchased for $20,000 decreases at a rate of 12% per year. What will be the value of the car after 3 years?

1) $12,800.00
2) $13,629.44
3) $17,600.00
4) $28,098.56

Answer: 2) $13,629.44

Depreciation lowers the price. It is exponential decay. So Choice (4) can be eliminated immediately.

If you don't know how to do compounded interest, you can evaluate simple interest three times, as long as you subtract the value you find the first time before finding interest for the next year.

As one equation, it's A = 20000(1 - .12)³ = 13629.44

If you got Choice (1), you subtracted 36% using simple interest. This is incorrect.

If you get Choice (3), you found the value after a single year.

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25. For which set of values of x is the algebraic expression (x² - 16) / (x² - 4x - 12) undefined?

1) {-6, 2}
2) {-4, 3}
3) {-4, 4}
4) {-2, 6}

Answer: 4) {-2, 6}

A fraction is undefined when the denominator equals 0. Set the denominator equal to 0 and solve for x. (You can also work backward from the choices.)

x² - 4x - 12 = 0

(x - 6)(x + 2) = 0

x - 6 = 0 or x + 2 = 0

x = 6 or x = -2

More to come. Comments and questions welcome.

More Regents problems.

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