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 IV: Each correct answer will receive 4 credits. Partial credit is available.


37. Solve algebraically for x:

3 / 4 = -(x + 11) / (4x) + 1 / (2x)

Answer:

You can multiply the entire equation by 4x to eliminate the fractions, or you can multiply the final fraction by (2/2), which would allow you to combine the fractions, which allows you to cross-multiply to solve the proportion.

Method 1:

(4x) [ 3 / 4 = -(x + 11) / (4x) + 1 / (2x) ]

3x = -(x + 11) + 2

3x = -x - 11 + 2

3x = -x - 9

4x = - 9

x = -9/4

Method 2:

3 / 4 = -(x + 11) / (4x) + [1 / (2x)][2/2]

3 / 4 = -(x + 11) / (4x) + 2 / (4x)

3 / 4 = (-x - 11) / (4x) + 2 / (4x)

3 / 4 = (-x - 9) / (4x)

12x = -4x - 36

16x = -36

x = -36/16 = -9/4

Note: only do one of the above. If you solve it twice, both will be graded and you will get the average of the two of them, not the better one.

38.An outfit Jennifer wears to school consists of a top, a bottom, and shoes. Possible choices are listed below.

Tops: T-shirt, blouse, sweater

Bottoms: jeans, skirt, capris

Shoes: flip-flops, sneakers

List the sample space or draw a tree diagram to represent all possible outfits consisting of one type of top, one type of bottom, and one pair of shoes.

Determine how many different outfits contain jeans and flip-flops.

Determine how many different outfits do not include a sweater.

Answer:

A sample space is a listing of ALL the possible combinations. A tree diagram should be smaller. Note that there are 3 * 3 * 2 = 18 possibilities. You can abbreviate, using just the first letter, except that you should use Sw for sweater, Sk for skirt, and Sn for sneakers to avoid confusion -- even if you think it's "obvious".

The sample space would be:

T J F, T J Sn, T Sk F, T Sk Sn, T C F, T C Sn

B J F, B J Sn, B Sk F, B Sk Sn, B C F, B C Sn

Sw J F, Sw J Sn, Sw Sk F, Sw Sk Sn, Sw C F, Sw C Sn

A tree diagram would look like this:

The number of outfits containing jeans and flip-flops is 3. You don't need to provide work IF you answered the first part of the problem. If you did NOT make a tree diagram or sample space then you MUST list the three outfits here to get the credit.

Twelve outfits do not include a sweater. Again, the tree diagram/sample space is the work for this question. If you didn't answer it, you need to explain how you got your answer, such as using the Counting Principle.

39. Solve the following system of inequalities graphically on the set of axes below.

3x + y < 7

y ≥ 2/3 x - 4

State the coordinates of a point in the solution set.

Answer:

Do NOT pick the intersection point or any point on the dashed line as a point in the solution. Nothing on the dashed line, including the intersection of the two lines, is a solution.

You can rewrite the first equation as y < -3x + 7 to put it in your graphing calculator.

Label at least one line (with the original inequality). Both is better.

Label any point in the double-shaded area.

You're graph should look like the one below. Point (0,0), for example, is a point in the solution set.

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

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