Math 009 Quiz 4 Fall 2016 Professor Dr Kate Bauer Name

Math 009 Quiz 4 Fall 2016 Professor Dr Kate Bauer Name

The quiz is worth 50 points. There are 10 problems, each worth 5 points. Your score will be converted to a percentage and posted in your assignment folder with comments. Open book and open notes are allowed, and you may take as long as needed, provided you submit by the deadline. You may refer to your textbook, notes, and online materials but not consult others. Show all work to receive full credit; incomplete work may earn partial or no credit. You can type your work, create a document, or scan it, but include your name. Review submission instructions in the Quizzes Module. For questions, contact the instructor via email. Read the instructions for showing work posted in the LEO classroom before starting. You must include a signed statement at the end affirming you completed the quiz independently, or you'll receive a zero.

Paper For Above instruction

This paper addresses a set of algebraic problems designed to assess understanding of geometric reasoning, algebraic equations, inequalities, and basic finance applications. Each question is examined in detail, demonstrating problem-solving techniques, algebraic manipulations, and relevant interpretations necessary for a comprehensive solution.

Question 1: Angles in a Triangle

Tanya's triangular garden bed has angles where the largest angle is 30° less than twice the smallest, and the middle angle is 10° more than the smallest. To find the measures of all three angles, denote the smallest angle as x degrees. Then, the largest angle is 2x - 30 degrees, and the middle angle is x + 10 degrees. Since the sum of angles in a triangle is 180°, set up the equation:

x + (2x - 30) + (x + 10) = 180

Simplify and solve for x:

4x - 20 = 180

4x = 200

x = 50°

Therefore, the angles are:

  • Smallest angle: 50°
  • Middle angle: 50 + 10 = 60°
  • Largest angle: 2(50) - 30 = 70°

Hence, the measures of the three angles are 50°, 60°, and 70° respectively, satisfying the triangle angle sum.

Question 2: Distance from Starting Point

Jacob's 3420-mile bicycle trip involves a moment where his distance from the start is exactly 60 miles more than twice the remaining distance to the endpoint. If d represents the distance from his start point at that moment, the remaining distance is 3420 - d. According to the problem:

d = 2(3420 - d) + 60

Expand and rearrange:

d = 6840 - 2d + 60

d + 2d = 6900

3d = 6900

d = 2300 miles

Jacob was 2300 miles from the starting point when he made that calculation. The remaining distance to the finish line was:

3420 - 2300 = 1120 miles

Thus, he was 1120 miles from Portland when he calculated that distance.

Question 3: Solving Inequality 3

Given the inequality:

3(7x - 4) > 0

First, expand:

21x - 12 > 0

Add 12 to both sides:

21x > 12

Divide both sides by 21:

x > 12/21 = 4/7

Set-builder notation:

  • { x | x > 4/7 }

Interval notation:

  • ( 4/7, ∞ )

Graphically, this is the set of all real numbers greater than 4/7, represented on a number line as an open interval starting just after 4/7 extending to infinity.

Question 4: Solving Inequality 4

Given:

x - x/5

To clear fractions, multiply everything by the LCD, 5:

5(x) - 5(x/5)

5x - x

4x

x

Set-builder notation:

  • { x | x

Interval notation:

  • ( -∞, 5/2 )

The solution set includes all real numbers less than 2.5, to the left of 2.5 on the number line.

Question 5: Inequality with Fractions 2

Given:

(x - 1)/2 ≥ (x + 3)/4

Multiply through by 4 (LCD) to clear denominators:

2( x - 1 ) ≥ ( x + 3 )

2x - 2 ≥ x + 3

Subtract x from both sides:

x - 2 ≥ 3

x ≥ 5

Set-builder notation:

  • { x | x ≥ 5 }

Interval notation:

  • [5, ∞)

Graphically, includes all points from 5 onward to infinity.

Question 6: Inequality with Fractions 3

Given:

(4x - 2)/3 ≤ (7x + 5)/6

Multiply both sides by 6 (LCD):

2(4x - 2) ≤ 7x + 5

8x - 4 ≤ 7x + 5

Subtract 7x from both sides:

x - 4 ≤ 5

x ≤ 9

Set-builder notation:

  • { x | x ≤ 9 }

Interval notation:

  • ( -∞, 9 ]

Question 7: Inequality x + (x + 2)

Combine like terms:

2x + 2

Subtract 2 from both sides:

2x

Divide by 2:

x

Set-builder notation:

  • { x | x

Interval notation:

  • ( -∞, 3/2 )

Question 8: Investment Interest Problem

Melissa plans to invest a total of $17,000 in two accounts. One is a CD with 12% annual simple interest on $10,000. The remaining amount, $7,000, must earn an interest rate r to ensure total yearly interest is at least $1,900. The total interest from both investments is:

0.12(10,000) + r(7,000) ≥ 1,900

Simplify:

1,200 + 7,000r ≥ 1,900

Subtract 1,200:

7,000r ≥ 700

Divide by 7,000:

r ≥ 700 / 7,000 = 0.10 or 10%

Therefore, the remainder must earn at least 10% interest.

Set-builder notation:

  • { r | r ≥ 0.10 }

Interval notation:

  • [0.10, ∞)

Question 9: Solving for y in terms of x

Given the equation:

3y = 2x + 6

Solve for y:

y = (2x + 6)/3

Generate four solutions by choosing values for x:

  • x = 0: y = (0 + 6)/3 = 2
  • x = 3: y = (6 + 6)/3 = 4
  • x = -3: y = (-6 + 6)/3 = 0
  • x = 6: y = (12 + 6)/3 = 6

The ordered pairs are:

  • (0, 2)
  • (3, 4)
  • (-3, 0)
  • (6, 6)

Question 10: Plotting Points and Line

Plot the points (0, 2), (3, 4), (-3, 0), and (6, 6) on a coordinate grid. Connect them with a straight line. Verify that all points lie on the line y = (2x + 6)/3 and that the line accurately represents the equation. This confirms the points are correct solutions to the equation, illustrating a linear relationship between x and y.

References

  • Stewart, J. (2015). Precalculus: Mathematics for Calculus. Cengage Learning.
  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
  • Blitzer, R. (2014). Algebra and Trigonometry. Pearson.
  • Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Early Transcendentals. Wiley.
  • Baratto, M., Bergman, D., & Cioffi, M. (2017). Beginning Algebra. Pearson.
  • Hartshorne, R. (2013). Geometry: Euclid and Beyond. Springer.
  • Knuth, D. (1997). The Art of Computer Programming. Addison-Wesley.
  • Fitzpatrick, R. (2012). Mathematics for Economics and Finance. Cambridge University Press.
  • Schmidt, P. (2014). Statistics with R. Springer.
  • Briggs, W. L., Cochran, L. B., & Gillett, W. (2012). Calculus. Pearson.

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