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

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

More Regents problems.

Integrated Algebra Regents, January 2012

Part III: Each correct answer will receive 3 credits. Partial credit is available.


34. Solve algebraically for: 2(x - 4) > 1/2(5 - 3x)

Answer:

You can avoid dealing with the fraction by multiplying the inequality by 2 before you use the Distributive Property.

2(x - 4) > 1/2(5 - 3x)
4(x - 4) > 5 - 3x
4x - 16 > 5 - 3x
7x - 16 > 5
7x > 21
x > 3

35. On the set of axes below, solve the following system of equations graphically. State the coordinates of the solution.

y = 4x + 1
2x + y = 5

Answer:

If you're using a graphing calculator to assist you, rewrite the second equation as y = -2x + 5.

The first line has a y-intercept of -1 and a slope of 4. The second line has a y-intercept of 5 and a slope of -2. Graph and label each line. Label the point of intersection and state the coordinates.

Your graph should like like the image below:

36. A turtle and a rabbit are in a race to see who is first to reach a point 100 feet away. The turtle travels at a constant speed of 20 feet per minute for the entire 100 feet. The rabbit travels at a constant speed of 40 feet per minute for the first 50 feet, stops for 3 minutes, and then continues at a constant speed of 40 feet per minute for the last 50 feet.
Determine which animal won the race and by how much time.

Answer:

The turtle travels 20 feet per minute, so it completes the race in (100 ft) / (20 ft/min) or 5 minutes.

The rabbit travels 40 ft/min for the first 50 feet. The time it takes is (50 ft) / (40 ft/min) = 1.25 minutes. It stops for 3 minutes, so it has travels 50 at the end of 4.25 minutes. It then repeats its pace for the first half of the race, which we already know is 1.25. A total of 100 feet takes the rabbit 4.5 minutes.

So the turtle won by a half a minute. You must state both of those facts for full credit.

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

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