Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


16. Which expression represents (x² - 3x - 10) / (x² - 25) in simplest form?

1) 2 / 5
2) (x + 2) / (x + 5)
3) (x - 2) / (x - 5)
4) (-3x - 10) / -25

Answer: 2) (x + 2) / (x + 5)

Factor both the numerator and the denominator into two binomials. Then cancel out the common factors that form a multiplicative identity. (That is N / N = 1)

(x² - 3x - 10) / (x² - 25)
( (x - 5) (x + 2) ) / ( (x - 5) (x + 5) )
(x + 2) / (x + 5)

You cannot cancel the x² terms in the original expression because they are not FACTORS. There are multiple terms being added and subtracted. Likewise, you cannot cancel the x terms in the final answer.

17. Which interval notation describes the set S = {x |1 < x < 10}?

1) [1,10]
2) (1,10]
3) [1,10)
4) (1,10)

Answer: 3) [1,10)

Interval notation is shorthand for a compound inequality that uses less than and less than or equal to. A rounded parenthesis means "less than" but not equal to, and a square bracket means use "less than or equal to". Note that Choice (4) is NOT a point on the coordinate plane, even though it looks like one. This is because mathematicians used to do everything by hand, so this notation saved a lot of space and ink.

The lower boundary has < so the left symbol will be a bracket. The upper boundary is <, so it gets a parenthesis.

18.The bull’s-eye of a dartboard has a radius of 2 inches and the entire board has a radius of 9 inches, as shown in the diagram below

If a dart is thrown and hits the board, what is the probability that the dart will land in the bull’s-eye?

1) 2/9
2) 7/9
3) 4/81
4) 49/81

Answer: 3) 4/81

Find probability by dividing the amount of positive results by the ammount all possible results. Wit a darboard, we need to compare the areas of the bull's-eye with the area of the entire board. That means that we have to find the Area of two circles and divide them.

Note that because we will divide them, we don't need to multiply by pi, because the two factors of pi will cancel.

Area of bull's-eye = (2)² * pi = 4 * pi

Area of dartboard = (9)² * pi = 81 * pi

(4 pi) / (81 pi) = 4 / 81

19. What is one-third of 3⁶?

1) 1²
2) 3²
3) 3⁵
4) 9⁶

Answer: 3) 3⁵

One-third of 3⁶ is 1/3 * 3⁶ = 3⁶ / 3 = 3⁶ / 3¹.

The rules of exponents say that you subtract the exponents: 6 - 1 = 5

The Axis of Symmetry is x = -b / (2a) = -24 / (2(-2)) = 6

20. The expression (2x + 13) / (2x + 6) - (3x - 6) / (2x + 6) is equivalent to

1) (-x + 19) / (2(x + 3))
2) (-x + 7) / (2(x + 3))
3) (5x + 19) / (2(x + 3))
4) (7x + 7) / (4x + 12)

Answer: 1) (-x + 19) / (2(x + 3))

You can only add fractions when the denominators are the same, and then the denominator stays the same. You do NOT add the denominators, so Choice (4) is eliminated.

(2x + 13) / (2x + 6) - (3x - 6) / (2x + 6)
( (2x + 13) - (3x - 6) ) / (2x + 6)
( 2x + 13 - 3x + 6) / (2x + 6)
( -x + 19 ) / (2x + 6)

More to come. Comments and questions welcome.

More Regents problems.