# Exam: 050291RR - Systems of Equations; Inequalities

Exam: 050291RR - Systems of Equations; Inequalities

1. Solve the system of equations x - 4y = –8 and –3x + 12y = 24.

A. There is one solution, and it is (–4, 1).

B. There is no solution.

C. There are infinitely many solutions.

D. There is one solution, and it is (0, 2).

2. Aunt Jane's Pies had a tent at the county fair. Unfortunately their cash register broke, so they have no

receipts. They know from counting their left over paper plates that they made 413 sales. They know from

the cash box that they made $2,243. If they only sell two kinds of items at the fair tent, a piece of pie for

$4 and pie á là mode for $7, help them figure out how many of each kind they sold.

A. They sold 610 pieces of pie and 4683 pies á là mode.

B. They sold 216 pieces of pie and 197 pies á là mode.

C. The system of equations is inconsistent, and therefore their plate counting or money counting must have an error.

D. They sold 355 pieces of pie and 58 pies á là mode.

3. Solve the inequality . Give the result in set notation and graph it.

4. When solving a system of equations using Cramer's Rule, if D

x

= 0, D

y

= –1, D

= 1, and D = 0, then

what can you conclude?

A. The system is inconsistent.

B. The system has one solution, (0, 0, 0).

C. The system has one solution, (0, –1, 1).

D. The system is dependent.

5. Graph the following solution set:

x + y = 4

x = 0

y = 0

6. Are the two equations –6 + y = 2x and 2y - 4x = 12 dependent?

A. Yes, because they have the same graph.

B. No, because they are not parallel.

C. No, because the equations are not written the same.

D. Yes, because both are the equations of straight lines.

7. Solve the system of equations 2x - 2y - 2z = 3, x + 4y - z = 2, and –2x - 8y + 2z = –4.

A. There is one solution, (0.1, 0.1, –1.5).

B. There are infinitely many solutions, of the form (x, 0.1, x -1.6).

C. There are infinitely many solutions, of the form (0.1, 0.1, –1.5).

D. There is no solution.

8. What is the formula for the determinant of a 3 x 3 matrix ?

A.

B.

C.

D.

9. Find the value of the expression –|–18|.

A. 18

B. 0

C. Undefined

D. –18

10. If the edge isn't included in the graph of an inequality, you should draw it as a/an _______ line.

A. open

B. solid

C. closed

D. dashed

11. Find the equation of the boundary line in the graph below. Then give the inequality represented by the

shaded area.

A.

B.

C.

D.

12. Solve the equation |6x + 3| = 15

A. x = 4

B. x = –3

C. x = 2

D. x = –3 or x = 2

13. The matrix below is the augmented matrix of a system of three equations in the variables x and y. Solve

for x and y.

A. One solution,

B. One solution (0, 9)

C. Infinitely many solutions

D. No solution

14. Graph the following solution set:

x = y

y = x

2

A.

B.

C.

D.

15. Solve the equation |x| = 7.

A. x = –7

B. Undefined

C. x = 7

D. x = 7 or x = –7

16. When solving the system of equations 6x + 2y = –1 and –x + 10y = 5 by Cramer's Rule, what are the

values of D, D

A. D = 62, D

B. D = 58, D

x

x

x

, and D

= –20, D

= –20, D

y

y

y

?

= 29

= –29

C. D = 58, D

x

= 20, D

= –28

D. D = 62, D

x

= 20, D

y

= 28

y

17. Solve the inequality |5x + 10| = 15. Write the solution in interval notation and graph it.

A.

B.

C.

D.

18. Graph the following solution set:

y = x - 1

y = 2x

A.

B.

C.

D.

19. Which of the following phrases correctly describes the graph of the system of equations

and y = 2 - x?

A. The graph is of two lines that coincide.

B. The graph is of two lines that intersect at a single point.

C. The graph is of two parallel lines that do not intersect.

D. The graph is of a line and a parabola, which intersect at two points.

20. Consider two ships, one on a course described by the equation 0.6x + 0.3y = 2.1 and the other on a

course described by the equation –0.3x + 0.1y = –1.8. Which of the following sentences best describes the

possibility of a collision?

A. There is a possibility of a collision at the point (5, –3) but a collision is not a certainty.

B. There is no possibility for a collision.

C. There is a possibility of a collision at the point (0, 7) but a collision is not a certainty.

D. There will certainty be a collision at the point (6, 0).

21. Solve the inequality . Give the result in set notation and graph it.

A.

B.

C.

D.

22. Graph the inequality y < 3x + 1.

A.

B.

C.

D.

23. Solve the inequality |2x - 4| < 10. Write the solution in interval notation and graph it.

A.

B.

C.

D.

24. Solve the inequality 4 < –z - 4 < 11. Give the result in set notation and graph it.

A.

B.

C.

D.

25. Graph the inequality 3x = –4y - 4.

Exam: 050294RR - Radicals and Rational Exponents

1. Compute the value of .

2. Rationalize the denominator of assuming x = 0 and y = 0.

3. What are the mean and standard deviation of the data – 6, 12, 2, – 4, 1, 6, 0, 3?

A. The mean is and the standard deviation is approximately 5.6252.

B. The mean is 4.86 and the standard deviation is 5.63.

C. The mean is and the standard deviation is approximately 5.6252.

D. The mean is and the standard deviation is 5.

4. Expand and simplify. Assume y = 0.

5. Which of the following best describes imaginary numbers?

A. They are not numbers, but are useful in solving equations.

B. They consist of two values, the principal imaginary number i and its negative –i.

C. They are the values of expressions of the form for various real numbers a, as long as a ≠ 0.

D. They are the complex numbers.

6. Which of these points is 5 units away from the point (6, –1)?

7. Which expression has the same value as 25½?

8. Two children in nearby houses attempt to use walkie-talkies to communicate. The walkie-talkies reach

one quarter of a mile (1320 feet). From one child's house to the other, the walk along the city sidewalks is

as follows: Proceed 450 feet from the first house to the nearest corner, turn right 90° and proceed another

1050 feet. Are the children's houses within the 1320-foot range of one another? Choose the best answer.

A. No, but if the turn were to the left instead, they would be within range.

B. No, because the distance is greater than 1320 feet.

C. Yes, because city blocks are much smaller than one quarter of a mile.

D. Yes, as the distance formula indicates.

9. Simplify assuming the variables represent non-negative numbers.

10. Simplify (6 - i)(2 + i).

A. 11 + 8i

B. 11 + 4i

C. 13 + 4i

D. 13 - 8i

11. Which of these expressions simplifies to ?

12. To solve for x, begin with which of these steps?

A. Eliminate the negative in the second radical expression.

B. Combine the two like radicals, then square both sides.

C. Isolate one radical expression.

D. Square both sides of the equation.

13. Which of these expressions is in simplified form?

14. Simplify by rationalizing the denominator.

15. Simplify .

16. Combine like radicals in .

17. Solve for x.

A. The two solutions for are complex numbers.

B. x = 4 or x = –4

C. x = 2 or x = 8

D. x = 2

18. Which of these phrases best describes the standard deviation?

A. It is a measure of variability.

B. It is equal to the mean squared.

C. It is a radical expression using n variables.

D. It increases as more measurements are taken.

19. Simplify .

20. Solve for x.

A. 40

B. 4

C. 50

D. –41

21. Simplify .

22. Which of these radical expressions simplifies to x?

23. Choose the best description of the radical expression .

A. It can be simplified to –3.

B. It can be simplified, but the result is a complex number.

C. It can be simplified to , but no further.

D. It is in simplified form.

24. If the hypotenuse of a right triangle is 6m and one side is 4m, what is the length of the other side?

25. Expand and simplify. Assume c = 0 and d = 0.

Exam: 050293RR - Rational Expressions

1. Find the result of the polynomial division (3x

2. Perform the indicated operations and simplify the following expression:

3. Solve the following equation:

A. z = –4

B. z = –6

C. z = 4

D. z = 6

4. Simplify the following expression:

5. Solve this proportion for the variable:

A. y = –2

B. y = 4

C. y = –4

D. y = 2

6. Solve the following equation:

A. y = –4

B. y = –3

C. y = 3

D. y = 4

7. Perform the indicated operations and simplify the following expression:

8. Express this sentence as a formula: z varies directly with the square of s.

9. Factor and simplify .

A. It is already fully simplified.

10. Simplify the expression

11. Your textbook says that the focal length f of a lens is given by the lensmaker's formula where r

A.

B.

C. 4

D. 16.67

12. Solve for k in .

A. k = –1

B. k = –1 and k = 2

C. k = 3

D. k = –1 and k = 3

13. Simplify the following expression:

14. It costs $500 to have a booth at the fair for one day and $2.75 for Aunt Ida to produce one of her famous shepherd's pies. What is the function that gives the average cost in dollars per pie for one of Aunt Ida's days at the fair?

15. Which sum yields ?

16. The total resistance R of two resistors in parallel is given by where R

first and R

2 the resistance of the second. Solve that equation for R

17. Solve the following equation:

A. t = 1

B. { }

1

and simplify.

1

is the resistance of the

C. t = 2

D. t = –1

18. Simplify the expression

19. The intensity I of light varies inversely with the square of the distance d from the source, expressed in the equation . If the intensity is of a candela 16 feet from the source, what is the constant of proportionality?

A. 256 candela-square-feet

B. 128 candela-square-feet

C. 32 candela-square-feet

D. 2 candela-square-feet

20. Perform the indicated operations and simplify the following expression:

1. Solve the system of equations x - 4y = –8 and –3x + 12y = 24.

A. There is one solution, and it is (–4, 1).

B. There is no solution.

C. There are infinitely many solutions.

D. There is one solution, and it is (0, 2).

2. Aunt Jane's Pies had a tent at the county fair. Unfortunately their cash register broke, so they have no

receipts. They know from counting their left over paper plates that they made 413 sales. They know from

the cash box that they made $2,243. If they only sell two kinds of items at the fair tent, a piece of pie for

$4 and pie á là mode for $7, help them figure out how many of each kind they sold.

A. They sold 610 pieces of pie and 4683 pies á là mode.

B. They sold 216 pieces of pie and 197 pies á là mode.

C. The system of equations is inconsistent, and therefore their plate counting or money counting must have an error.

D. They sold 355 pieces of pie and 58 pies á là mode.

3. Solve the inequality . Give the result in set notation and graph it.

4. When solving a system of equations using Cramer's Rule, if D

x

= 0, D

y

= –1, D

= 1, and D = 0, then

what can you conclude?

A. The system is inconsistent.

B. The system has one solution, (0, 0, 0).

C. The system has one solution, (0, –1, 1).

D. The system is dependent.

5. Graph the following solution set:

x + y = 4

x = 0

y = 0

6. Are the two equations –6 + y = 2x and 2y - 4x = 12 dependent?

A. Yes, because they have the same graph.

B. No, because they are not parallel.

C. No, because the equations are not written the same.

D. Yes, because both are the equations of straight lines.

7. Solve the system of equations 2x - 2y - 2z = 3, x + 4y - z = 2, and –2x - 8y + 2z = –4.

A. There is one solution, (0.1, 0.1, –1.5).

B. There are infinitely many solutions, of the form (x, 0.1, x -1.6).

C. There are infinitely many solutions, of the form (0.1, 0.1, –1.5).

D. There is no solution.

8. What is the formula for the determinant of a 3 x 3 matrix ?

A.

B.

C.

D.

9. Find the value of the expression –|–18|.

A. 18

B. 0

C. Undefined

D. –18

10. If the edge isn't included in the graph of an inequality, you should draw it as a/an _______ line.

A. open

B. solid

C. closed

D. dashed

11. Find the equation of the boundary line in the graph below. Then give the inequality represented by the

shaded area.

A.

B.

C.

D.

12. Solve the equation |6x + 3| = 15

A. x = 4

B. x = –3

C. x = 2

D. x = –3 or x = 2

13. The matrix below is the augmented matrix of a system of three equations in the variables x and y. Solve

for x and y.

A. One solution,

B. One solution (0, 9)

C. Infinitely many solutions

D. No solution

14. Graph the following solution set:

x = y

y = x

2

A.

B.

C.

D.

15. Solve the equation |x| = 7.

A. x = –7

B. Undefined

C. x = 7

D. x = 7 or x = –7

16. When solving the system of equations 6x + 2y = –1 and –x + 10y = 5 by Cramer's Rule, what are the

values of D, D

A. D = 62, D

B. D = 58, D

x

x

x

, and D

= –20, D

= –20, D

y

y

y

?

= 29

= –29

C. D = 58, D

x

= 20, D

= –28

D. D = 62, D

x

= 20, D

y

= 28

y

17. Solve the inequality |5x + 10| = 15. Write the solution in interval notation and graph it.

A.

B.

C.

D.

18. Graph the following solution set:

y = x - 1

y = 2x

A.

B.

C.

D.

19. Which of the following phrases correctly describes the graph of the system of equations

and y = 2 - x?

A. The graph is of two lines that coincide.

B. The graph is of two lines that intersect at a single point.

C. The graph is of two parallel lines that do not intersect.

D. The graph is of a line and a parabola, which intersect at two points.

20. Consider two ships, one on a course described by the equation 0.6x + 0.3y = 2.1 and the other on a

course described by the equation –0.3x + 0.1y = –1.8. Which of the following sentences best describes the

possibility of a collision?

A. There is a possibility of a collision at the point (5, –3) but a collision is not a certainty.

B. There is no possibility for a collision.

C. There is a possibility of a collision at the point (0, 7) but a collision is not a certainty.

D. There will certainty be a collision at the point (6, 0).

21. Solve the inequality . Give the result in set notation and graph it.

A.

B.

C.

D.

22. Graph the inequality y < 3x + 1.

A.

B.

C.

D.

23. Solve the inequality |2x - 4| < 10. Write the solution in interval notation and graph it.

A.

B.

C.

D.

24. Solve the inequality 4 < –z - 4 < 11. Give the result in set notation and graph it.

A.

B.

C.

D.

25. Graph the inequality 3x = –4y - 4.

Exam: 050294RR - Radicals and Rational Exponents

1. Compute the value of .

2. Rationalize the denominator of assuming x = 0 and y = 0.

3. What are the mean and standard deviation of the data – 6, 12, 2, – 4, 1, 6, 0, 3?

A. The mean is and the standard deviation is approximately 5.6252.

B. The mean is 4.86 and the standard deviation is 5.63.

C. The mean is and the standard deviation is approximately 5.6252.

D. The mean is and the standard deviation is 5.

4. Expand and simplify. Assume y = 0.

5. Which of the following best describes imaginary numbers?

A. They are not numbers, but are useful in solving equations.

B. They consist of two values, the principal imaginary number i and its negative –i.

C. They are the values of expressions of the form for various real numbers a, as long as a ≠ 0.

D. They are the complex numbers.

6. Which of these points is 5 units away from the point (6, –1)?

7. Which expression has the same value as 25½?

8. Two children in nearby houses attempt to use walkie-talkies to communicate. The walkie-talkies reach

one quarter of a mile (1320 feet). From one child's house to the other, the walk along the city sidewalks is

as follows: Proceed 450 feet from the first house to the nearest corner, turn right 90° and proceed another

1050 feet. Are the children's houses within the 1320-foot range of one another? Choose the best answer.

A. No, but if the turn were to the left instead, they would be within range.

B. No, because the distance is greater than 1320 feet.

C. Yes, because city blocks are much smaller than one quarter of a mile.

D. Yes, as the distance formula indicates.

9. Simplify assuming the variables represent non-negative numbers.

10. Simplify (6 - i)(2 + i).

A. 11 + 8i

B. 11 + 4i

C. 13 + 4i

D. 13 - 8i

11. Which of these expressions simplifies to ?

12. To solve for x, begin with which of these steps?

A. Eliminate the negative in the second radical expression.

B. Combine the two like radicals, then square both sides.

C. Isolate one radical expression.

D. Square both sides of the equation.

13. Which of these expressions is in simplified form?

14. Simplify by rationalizing the denominator.

15. Simplify .

16. Combine like radicals in .

17. Solve for x.

A. The two solutions for are complex numbers.

B. x = 4 or x = –4

C. x = 2 or x = 8

D. x = 2

18. Which of these phrases best describes the standard deviation?

A. It is a measure of variability.

B. It is equal to the mean squared.

C. It is a radical expression using n variables.

D. It increases as more measurements are taken.

19. Simplify .

20. Solve for x.

A. 40

B. 4

C. 50

D. –41

21. Simplify .

22. Which of these radical expressions simplifies to x?

23. Choose the best description of the radical expression .

A. It can be simplified to –3.

B. It can be simplified, but the result is a complex number.

C. It can be simplified to , but no further.

D. It is in simplified form.

24. If the hypotenuse of a right triangle is 6m and one side is 4m, what is the length of the other side?

25. Expand and simplify. Assume c = 0 and d = 0.

Exam: 050293RR - Rational Expressions

1. Find the result of the polynomial division (3x

2. Perform the indicated operations and simplify the following expression:

3. Solve the following equation:

A. z = –4

B. z = –6

C. z = 4

D. z = 6

4. Simplify the following expression:

5. Solve this proportion for the variable:

A. y = –2

B. y = 4

C. y = –4

D. y = 2

6. Solve the following equation:

A. y = –4

B. y = –3

C. y = 3

D. y = 4

7. Perform the indicated operations and simplify the following expression:

8. Express this sentence as a formula: z varies directly with the square of s.

9. Factor and simplify .

A. It is already fully simplified.

10. Simplify the expression

11. Your textbook says that the focal length f of a lens is given by the lensmaker's formula where r

A.

B.

C. 4

D. 16.67

12. Solve for k in .

A. k = –1

B. k = –1 and k = 2

C. k = 3

D. k = –1 and k = 3

13. Simplify the following expression:

14. It costs $500 to have a booth at the fair for one day and $2.75 for Aunt Ida to produce one of her famous shepherd's pies. What is the function that gives the average cost in dollars per pie for one of Aunt Ida's days at the fair?

15. Which sum yields ?

16. The total resistance R of two resistors in parallel is given by where R

first and R

2 the resistance of the second. Solve that equation for R

17. Solve the following equation:

A. t = 1

B. { }

1

and simplify.

1

is the resistance of the

C. t = 2

D. t = –1

18. Simplify the expression

19. The intensity I of light varies inversely with the square of the distance d from the source, expressed in the equation . If the intensity is of a candela 16 feet from the source, what is the constant of proportionality?

A. 256 candela-square-feet

B. 128 candela-square-feet

C. 32 candela-square-feet

D. 2 candela-square-feet

20. Perform the indicated operations and simplify the following expression:

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