# Systems of Equations

Systems of Equations; Inequalities

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

A. x = –3 or x = 2

B. x = 2

C. x = –3

D. x = 4

2. When solving a system of equations using Cramer's Rule, if Dx = 0, Dy = –1, Dz = 1, and D = 0, then what can you conclude?

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

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

C. The system is dependent.

D. The system is inconsistent.

Solve the inequality –2 (3 + x) < 4x + 4 < 8x. Give the result in set notation and graph it.

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

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

B. There is no solution.

C. There are infinitely many solutions, of the form (x, 0.1, x −1.6).

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

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

A. closed

B. dashed

C. open

D. solid

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

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

8. Solve the equation |x| = 7.

A. x = 7 or x = –7

B. x = 7

C. Undefined

D. x = –7

9. Graph the following solution set:

x ≤ y2

y ≥ x

10. 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 355 pieces of pie and 58 pies á là mode.

B. They sold 610 pieces of pie and 4683 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 216 pieces of pie and 197 pies á là mode.

11. Graph the inequality y ≥ –3.

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

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

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

C. There are infinitely many solutions.

D. There is no solution.

13. Choose the correct ways to fill in the blanks in the following sentence.

To solve a system of equations using the matrix method, use __________ to transform the augmented matrix into one with __________, then proceed to back-substitute.

A. elementary row operations, zeros below the diagonal

B. the coefficient matrix, Gaussian elimination

C. the coefficient matrix, an inverse

D. multiplication and addition, zeros in its final column

14. Find the equation of the boundary line in the graph below. Then give the inequality represented by the shaded area.

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

16. 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.

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

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

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 intersect at a single point.

B. The graph is of two lines that coincide.

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. Graph the following solution set:

y ≤ x − 1

y ≥ 2x

21. Graph the solution set of

x + 2y ≤ 3

x + 2y ≤ 4

22. When solving the system of equations 6x + 2y = –1 and –x + 10y = 5 by Cramer's Rule, what are the values of D, Dx, and Dy?

A. D = 58, Dx = –20, Dy = –29

B. D = 58, Dx = 20, Dy = –28

C. D = 62, Dx = 20, Dy = 28

D. D = 62, Dx = –20, Dy = 29

23. Solve the system of equations 2x − y + z = –7, x − 3y + 4z = –19, and –x + 4y − 3z = 18.

A. There is one solution, (1, 2, 3).

B. There is one solution, (–1, 2, –3).

C. There is one solution, (1, –2, 3).

D. There is one solution, (–1, –2, –3).

24. Graph the inequality 3x ≤ –4y − 4.

25. 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. Infinitely many solutions

B. One solution (0, 9)

C. One solution,

D. No solution

Radicals and Rational Exponents

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

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

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

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

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

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

A. Isolate one radical expression.

B. Square both sides of the equation.

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

D. Eliminate the negative in the second radical expression.

3. Solve for x.

4. What best describes the triangle whose corners are located at the points (1, 1), (2, 2), and (0, 16)?

A. It is a right triangle.

B. None of the above.

C. It has area 22 square units.

D. It is isosceles.

5. Which of these expressions is in simplified form?

6. Simplify .

7. Which of the following best describes imaginary numbers?

A. They are the complex numbers.

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

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

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

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

Simplify assuming the variables represent non-negative numbers.

10. Simplify .

11. Expand and simplify. Assume c ≥ 0 and d ≥ 0.

12. Combine like radicals in .

13. Simplify .

14. 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. Yes, as the distance formula indicates.

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

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

15. Simplify .

16. Which of these expressions simplifies to ?

17. Which expression has the same value as 25½?

18. Simplify i23.

A. i

B. –i

C. 1

D. –1

19. Simplify (6 − i)(2 + i).

A. 13 + 4i

B. 11 + 4i

C. 13 − 8i

D. 11 + 8i

20. Which of these radical expressions simplifies to x?

21. Rationalize the denominator of assuming x ≥ 0 and y ≥ 0.

22. Simplify by rationalizing the denominator.

23. Solve for x.

A. x = 2 or x = 8

B. The two solutions for are complex numbers.

C. x = 4 or x = –4

D. x = 2

24. Solve for x.

A. 50

B. 40

C. 4

D. –41

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

A. It increases as more measurements are taken.

B. It is a radical expression using n variables.

C. It is equal to the mean squared.

D. It is a measure of variability.

Polynomials and Polynomial Functions

Solve the equation

2. Completely factor the expression 7(x − y) − z(x − y).

A. (x − y)(7 + z)

B. Prime

C. (x − y)2(7 − z)3

D. (x − y)(7 − z)

3. Completely factor the expression 16x4 − 81y4.

A. (4x2 − 9y2)(4x2 + 9y2)

B. 0

C. (2x + 3y)(2x − 3y)(4x2 + 9y2)

D. Prime

4. Perform the indicated operations on the expression below.

(5a3 + 3a - 2) - (4a3 + a2 + 5)

A. 20a3 + 3a2 − 10

B. a3 − a2 + 3a − 7

C. 3a6 − 7

D. a3 + a2 + 3

5. Find the product of –3a3b(2a0b4 − 4a2b3)

A. –6b4 + 12a6b3

B. 12a5b4 − 6a3b5

C. 14a5b4 − 6b5

D. –6b5 + 14a5b4

6. Completely factor the expression r2 − 2r + 1.

A. (r + 1)(r − 1)

B. (r − 1)(r − 1)

C. Prime

D. r(r − 2) + 1

7. Graph the polynomial function f(x) = x3 − 1. 8. Completely factor the expression y2 + 12y + 35.

A. (y + 7)(y + 5)

B. Prime

C. (y + 7)(y − 5)

D. (y − 7)(y − 5)

9. Completely factor the expression 2a3 − 128.

A. Prime

B. 2(a3 − 64)

C. 2(a − 4)3

D. 2(a − 4)(a2 + 4a + 16)

10. Solve the equation x2 + 4x − 45 = 0.

A. x = 9 and x = –5

B. x = –9 and x = 5

C. x = 15 and x = 30

D. No solution

11. Find the product of (2a + 3b)(2a − 3b).

A. 4a2 + 12ab − 9b2

B. 4a2 − 9b2

C. 2a2 − 3b2

D. 4a2 − 6b2

12. Completely factor the expression 4q2 + 27r4.

A. 27r4(4q2 + 1)

B. 4q(q + 27r)

C. Prime

D. 31q2r4

13. Find the product of (x − 2y)2.

A. x2 + 4y2

B. x2 + 2xy + 4y2

C. x2 + 4xy + 4y2

D. x2 − 4xy + 4y2

14. Completely factor the expression 16t3 − 50t2 + 36t.

A. Prime

B. 2t(8t − 25t + 18)

C. 2t(8t − 9)(t − 2)

D. 2t(8t + 9)(t + 2)

15. Find the value of the expression 2xy − x2y when x = 2 and y = 3.

A. –24

B. 6

C. 0

D. 24

16. Solve the equation 3z3 − 300z = 0.

A. z = 0

B. z = –4 and z = 1

C. Prime

D. z = 0, z = 10, and z = –10

17. Completely factor the expression a2 + 4b − ab − 4a.

A. Prime

B. (a − b)(a − 4)

C. (a + b)(a + 4)

D. (a + b)(a − 4)

18. Completely factor the expression 48u4v4 − 18u2v2 − 3u8v5.

A. Nonfactorable

B. 3u2v2(16u2v2 − 6 − u6v3)

C. u2v2(48u2v2 − 18 − 3u6v3)

D. 3(16u4v4 − 6u2v2 − u8v5)

19. Simplify the expression 2ab4 − 3a2b2 − ab4 + a2b2.

A. a2b8 − a2b8 − 2a4b4

B. 0

C. 2a2b8 − 3a4b4

D. ab4 − 2a2b2

20. Simplify the expression –2(4y2 + 3z3 + 5) + 3(2y2 − 5z3 + 3).

A. 9z3 − 14y2 + 19

B. –14y2z3 + 6y2 − 1

C. –21z3 − 2y2 − 1

D. 21z3 + 2y2 + 1

Rational Expressions

1. Simplify the following expression:

After t hours of work, Group A completes of a job and Group B completes of the same job. What expression represents the amount the two teams would complete after tseconds, working together?

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

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

5. Solve this proportion for the variable:

A. x = 7

B. x = 10

C. x = –7

D. x = –10

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

Simplify the expression

8. Solve the following equation:

Divide the following expression:

10. The total resistance R of two resistors in parallel is given by where R1 is the resistance of the first and R2 the resistance of the second. Solve that equation for R1 and simplify.

11 Solve this proportion for the variable:

12. Simplify .

13 Solve the following equation:

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

15. Factor and simplify .

16. Which sum yields ?

17. Simplify the following expression:

18. Simplify the expression

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

20. Divide the following expression:

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

A. x = –3 or x = 2

B. x = 2

C. x = –3

D. x = 4

2. When solving a system of equations using Cramer's Rule, if Dx = 0, Dy = –1, Dz = 1, and D = 0, then what can you conclude?

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

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

C. The system is dependent.

D. The system is inconsistent.

Solve the inequality –2 (3 + x) < 4x + 4 < 8x. Give the result in set notation and graph it.

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

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

B. There is no solution.

C. There are infinitely many solutions, of the form (x, 0.1, x −1.6).

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

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

A. closed

B. dashed

C. open

D. solid

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

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

8. Solve the equation |x| = 7.

A. x = 7 or x = –7

B. x = 7

C. Undefined

D. x = –7

9. Graph the following solution set:

x ≤ y2

y ≥ x

10. 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 355 pieces of pie and 58 pies á là mode.

B. They sold 610 pieces of pie and 4683 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 216 pieces of pie and 197 pies á là mode.

11. Graph the inequality y ≥ –3.

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

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

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

C. There are infinitely many solutions.

D. There is no solution.

13. Choose the correct ways to fill in the blanks in the following sentence.

To solve a system of equations using the matrix method, use __________ to transform the augmented matrix into one with __________, then proceed to back-substitute.

A. elementary row operations, zeros below the diagonal

B. the coefficient matrix, Gaussian elimination

C. the coefficient matrix, an inverse

D. multiplication and addition, zeros in its final column

14. Find the equation of the boundary line in the graph below. Then give the inequality represented by the shaded area.

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

16. 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.

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

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

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 intersect at a single point.

B. The graph is of two lines that coincide.

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. Graph the following solution set:

y ≤ x − 1

y ≥ 2x

21. Graph the solution set of

x + 2y ≤ 3

x + 2y ≤ 4

22. When solving the system of equations 6x + 2y = –1 and –x + 10y = 5 by Cramer's Rule, what are the values of D, Dx, and Dy?

A. D = 58, Dx = –20, Dy = –29

B. D = 58, Dx = 20, Dy = –28

C. D = 62, Dx = 20, Dy = 28

D. D = 62, Dx = –20, Dy = 29

23. Solve the system of equations 2x − y + z = –7, x − 3y + 4z = –19, and –x + 4y − 3z = 18.

A. There is one solution, (1, 2, 3).

B. There is one solution, (–1, 2, –3).

C. There is one solution, (1, –2, 3).

D. There is one solution, (–1, –2, –3).

24. Graph the inequality 3x ≤ –4y − 4.

25. 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. Infinitely many solutions

B. One solution (0, 9)

C. One solution,

D. No solution

Radicals and Rational Exponents

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

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

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

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

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

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

A. Isolate one radical expression.

B. Square both sides of the equation.

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

D. Eliminate the negative in the second radical expression.

3. Solve for x.

4. What best describes the triangle whose corners are located at the points (1, 1), (2, 2), and (0, 16)?

A. It is a right triangle.

B. None of the above.

C. It has area 22 square units.

D. It is isosceles.

5. Which of these expressions is in simplified form?

6. Simplify .

7. Which of the following best describes imaginary numbers?

A. They are the complex numbers.

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

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

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

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

Simplify assuming the variables represent non-negative numbers.

10. Simplify .

11. Expand and simplify. Assume c ≥ 0 and d ≥ 0.

12. Combine like radicals in .

13. Simplify .

14. 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. Yes, as the distance formula indicates.

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

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

15. Simplify .

16. Which of these expressions simplifies to ?

17. Which expression has the same value as 25½?

18. Simplify i23.

A. i

B. –i

C. 1

D. –1

19. Simplify (6 − i)(2 + i).

A. 13 + 4i

B. 11 + 4i

C. 13 − 8i

D. 11 + 8i

20. Which of these radical expressions simplifies to x?

21. Rationalize the denominator of assuming x ≥ 0 and y ≥ 0.

22. Simplify by rationalizing the denominator.

23. Solve for x.

A. x = 2 or x = 8

B. The two solutions for are complex numbers.

C. x = 4 or x = –4

D. x = 2

24. Solve for x.

A. 50

B. 40

C. 4

D. –41

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

A. It increases as more measurements are taken.

B. It is a radical expression using n variables.

C. It is equal to the mean squared.

D. It is a measure of variability.

Polynomials and Polynomial Functions

Solve the equation

2. Completely factor the expression 7(x − y) − z(x − y).

A. (x − y)(7 + z)

B. Prime

C. (x − y)2(7 − z)3

D. (x − y)(7 − z)

3. Completely factor the expression 16x4 − 81y4.

A. (4x2 − 9y2)(4x2 + 9y2)

B. 0

C. (2x + 3y)(2x − 3y)(4x2 + 9y2)

D. Prime

4. Perform the indicated operations on the expression below.

(5a3 + 3a - 2) - (4a3 + a2 + 5)

A. 20a3 + 3a2 − 10

B. a3 − a2 + 3a − 7

C. 3a6 − 7

D. a3 + a2 + 3

5. Find the product of –3a3b(2a0b4 − 4a2b3)

A. –6b4 + 12a6b3

B. 12a5b4 − 6a3b5

C. 14a5b4 − 6b5

D. –6b5 + 14a5b4

6. Completely factor the expression r2 − 2r + 1.

A. (r + 1)(r − 1)

B. (r − 1)(r − 1)

C. Prime

D. r(r − 2) + 1

7. Graph the polynomial function f(x) = x3 − 1. 8. Completely factor the expression y2 + 12y + 35.

A. (y + 7)(y + 5)

B. Prime

C. (y + 7)(y − 5)

D. (y − 7)(y − 5)

9. Completely factor the expression 2a3 − 128.

A. Prime

B. 2(a3 − 64)

C. 2(a − 4)3

D. 2(a − 4)(a2 + 4a + 16)

10. Solve the equation x2 + 4x − 45 = 0.

A. x = 9 and x = –5

B. x = –9 and x = 5

C. x = 15 and x = 30

D. No solution

11. Find the product of (2a + 3b)(2a − 3b).

A. 4a2 + 12ab − 9b2

B. 4a2 − 9b2

C. 2a2 − 3b2

D. 4a2 − 6b2

12. Completely factor the expression 4q2 + 27r4.

A. 27r4(4q2 + 1)

B. 4q(q + 27r)

C. Prime

D. 31q2r4

13. Find the product of (x − 2y)2.

A. x2 + 4y2

B. x2 + 2xy + 4y2

C. x2 + 4xy + 4y2

D. x2 − 4xy + 4y2

14. Completely factor the expression 16t3 − 50t2 + 36t.

A. Prime

B. 2t(8t − 25t + 18)

C. 2t(8t − 9)(t − 2)

D. 2t(8t + 9)(t + 2)

15. Find the value of the expression 2xy − x2y when x = 2 and y = 3.

A. –24

B. 6

C. 0

D. 24

16. Solve the equation 3z3 − 300z = 0.

A. z = 0

B. z = –4 and z = 1

C. Prime

D. z = 0, z = 10, and z = –10

17. Completely factor the expression a2 + 4b − ab − 4a.

A. Prime

B. (a − b)(a − 4)

C. (a + b)(a + 4)

D. (a + b)(a − 4)

18. Completely factor the expression 48u4v4 − 18u2v2 − 3u8v5.

A. Nonfactorable

B. 3u2v2(16u2v2 − 6 − u6v3)

C. u2v2(48u2v2 − 18 − 3u6v3)

D. 3(16u4v4 − 6u2v2 − u8v5)

19. Simplify the expression 2ab4 − 3a2b2 − ab4 + a2b2.

A. a2b8 − a2b8 − 2a4b4

B. 0

C. 2a2b8 − 3a4b4

D. ab4 − 2a2b2

20. Simplify the expression –2(4y2 + 3z3 + 5) + 3(2y2 − 5z3 + 3).

A. 9z3 − 14y2 + 19

B. –14y2z3 + 6y2 − 1

C. –21z3 − 2y2 − 1

D. 21z3 + 2y2 + 1

Rational Expressions

1. Simplify the following expression:

After t hours of work, Group A completes of a job and Group B completes of the same job. What expression represents the amount the two teams would complete after tseconds, working together?

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

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

5. Solve this proportion for the variable:

A. x = 7

B. x = 10

C. x = –7

D. x = –10

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

Simplify the expression

8. Solve the following equation:

Divide the following expression:

10. The total resistance R of two resistors in parallel is given by where R1 is the resistance of the first and R2 the resistance of the second. Solve that equation for R1 and simplify.

11 Solve this proportion for the variable:

12. Simplify .

13 Solve the following equation:

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

15. Factor and simplify .

16. Which sum yields ?

17. Simplify the following expression:

18. Simplify the expression

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

20. Divide the following expression:

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