Introduction to Set Notation

MAT135

In-Class Assignment

MAT135

In Class Assignments

Introduction to Set Notation

Introduction to Set Notation (continued)

1. List the elements in each set:

2. Find the union of the sets:

Introduction to Set Notation (continued)

3. Find the intersection of the sets:

4. Let .

Are the following statements true or false?

5. Write the following in set-builder notation:

Section 2.1 Solving Linear Equations

1. Solve the following equations:

2. The taxi meter was invented in 1891 by Wilhelm Bruhn. Chicago charges $1.80 plus $0.40 per mile for a taxi ride.

a) If the fare is $6.60, write an equation that shows how the fare is calculated.

b) How many miles were traveled if the fare was $6.60?

3. In 1992, twice as many people visited their doctor because of a cough than an earache. The total number of doctor’s visits for these two ailments was reported to be 45 million.

a) Let x represent the number of earaches reported in 1992. Write an expression for the number of coughs reported in 1992.

b) Write an equation that relates 45 million to the variable x.

c) How many people visited their doctor in 1992 to report an earache?

Section 2.2 Formulas and Functions

1. Solving for a Variable

Although you can’t print most of these out without signing up at the site, there are some useful presentations to help with this topic.

2. For each formula, express y as a function of x, then find y given that.

3. Solve for the indicated variable:

d. (Lateral area of a frustrum)

e. (Area of a trapezoid)

Section 2.3 Applications

Choose the equation that best describes the situation.

Section 2.3 Applications (continued)

10.

You may also want to view

Section 2.4 Inequalities

1. Solve the inequalities, graph each solution, and write the solution in interval notation:

2. A store selling art supplies finds that x sketch pads are sold each week at a price of p dollars each according to the formula x=900-300p. What price should they charge if they want to sell:

a) at least 300 pads each week?

b) fewer than 525 pads each week?

c) more than 600 pads each week?

d) at most 375 pads each week?

3. What about the solutions to these problems?

Section 2.5 Compound Inequalities

1. Solve the following inequalities. Use a line graph and interval notation to write each solution set.

2. A factory’s quality control department randomly selects a sample of 5 lightbulbs to test. In order to meet quality control standards, the lightbulbs in the sample must last an average of at least 950 hours. Four of the selected bulbs lasted 925 hours, 1000 hours, 950 hours, and 900 hours. How many hours must the fifth lightbulb last for the sample to meet quality control standards?

3. A worker earns $12 per hour plus $16 overtime pay for every hour over 40 hours. How many hours of overtime are needed to make between $600 and $800 per week?

4. Write each union or intersection as a single interval, if possible.

Section 2.6Absolute Value Equations and Inequalities

Some tutorials:

1. Solve: .

a. {-5}b.{3 }c.{-5 , 3}d. no solutions

2. Find all real solutions: .

a.{-2 }b.{-2 , 0}c.{0}d. no solutions

3. Find all real solutions:

a.b.c. d.

4. Find all real solutions to the equation. .

a. -4 b. no solutions c. 4 d. all real numbers

5. Solve the inequality: .

a. (, 7/3) b. (7/3 , ) c. (-7/3 , 7/3) d. (,)

6. Solve the inequality: .

a. [3 , ) b. ( , -1] U [3 , ) c. (- , -1] d. (- , )

Section 2.6 Absolute Value Equations and Inequalities

(continued)

7. Solve the inequality: .

a.b. ( , 0) U (16 , ) c. (0,16)d.

8. Find all real solutions to the equation with absolute value..

a. b.c.d.

9. Solve the inequality: .

a.b. ( , -2] U (6 , ) c.d.( , -6] U (2 , )

10. Solve the inequality: .

a.b. ( , -6) U (6 , ) c.d. ( , -6) U (0 , )

For more problems and answers:

Section 3.1 The Coordinate Plane

1. The x-axis on the graph below represents number of hours worked and the y-axis represents pay, in dollars.Approximately what is the pay for working 10, 20, 30, and 40 hours? Create a table that displays the data shown on the graph.

2. Find the x- and y-intercepts and graph the line.

3. Complete the ordered pairs so that the equation is satisfied.

Section 3.2 Slope of a Line

1. Find the slope of the line from the given graphs:

y y

2. Find the slope of the line through the given points. Then plot each pair of points and draw the line through them.

a) (2,1) (4,4) b) (2, -5) (3, -2) c) (1,5) (-4,-5)

3. A line, l, contains the points (3,4) and (-3,1). Give the slope of any line perpendicular to l.

4. Determine if the line through the points (7,2) and (-9,2) and the line through the points (4,-4) and (1, -4) are parallel, perpendicular, or neither.

5. Solve for y:

Section 3.3 Equation of a Line

1. What is the slope of the line whose equation is ?

[A]3/4 [B] 4/3 [C]3 [ D]

2. What is the equation of a line passing through the points (1,2) and (-2,5)?
[A]y = x + 3[B][C] [D]y = 3x + 3
3.Which of the following is the equation of a line with a slope of 0 and passing through the point (4,6)?
[A]x = 4 [B] [C]y = 6 [D]
4. What is the slope of a line passing through the points (3,5) and (-2,6)?
[A] [B] [C] [D]
5. A horizontal line has a slope of
[A] 0 [B] 1 [C] -1 [D] undefined
6. What is the slope of the line shown in the figure at the right?
[A] 4/3
[B] 3/4
[C]
[D]
7.What are the coordinates of the y-intercept of the equation y - 3x = 5?
[A] (0,3)[B] (0,-3) [C] (0,5) [D] (0,-5)
Section 3.3 Equation of a Line (continued)
8. The slope of a vertical line is
[A] 0 [B] 1 [C] [D] undefined
9.Find the slope of a line perpendicular to the line whose equation is 3y + 2x = 6.
[A] 2 [B] [C] [D] 3/2
10. Find the equation of the line parallel to the line whose equation is and whose
y-intercept is .
[A] [B] [C] [D]
11. Write an equation for a line passing through the points (c, 2b) and (c, 3b).
[A] [B] [C] [D]
12. Which is the equation of a line whose slope is undefined?
[A] [B][C]x = y [D]x + y = 0
13.What is the equation of the line shown in the graph at the right?
[A]
[B]y = 2x - 3
[C]
[D] /
14. Which of these equations represents a line parallel to the line ?
[A] [B] [C][D]
Section 3.3 Equation of a Line (continued)
15. Find the equation of the line that has a slope of and a y-intercept of .
[A] / [C]
[B] / [D]
Bottom of Form

16. Use the table at the right to answer the following questions:

a. Does Y1 pass through the origin?

b. What is the x-intercept of Y2?

c. For which values of x is Y1=Y2?

d. The graph of Y2 is in which quadrants?

e. The graph of Y1 is in which quadrants?

17. Write the general form of the equation of a line satisfying the given conditions.

a. b.

c. d.

18. Find the equation of a line with an x-intercept of (2,0) and a y-intercept of (0,3).

19. Complete the table:

Equation / Slope,m / x-intercept / y-intercept

Section 3.3 Equation of a Line (continued)

20. Match the graph with its description. You may assume that each tick mark represents one unit.

A.B.

C.D.

E.F.

1. This line has a slope of zero.

2. This line has a negative slope.

3. This line has an undefined slope.

4. This line is parallel to the line y = x.

5. This line has a positive y –intercept.

6. This line has a negative y –intercept.

Section 3.5 Functions and Relations

1.Determine whether the relation is a function. Identify the domain and the range.

d) e)

2. Let. Evaluate:

3. The function where V is value and t is time in years can be used to find the value of a large copy machine during the first 5 years of use.

a)What is the value of the copies after 3 years and 9 months?

b) What is the salvage value of the copier if it is replaced after 5 years?

c) State the domain of this function.

d) Sketch a graph of this function, clearly labeling values and the axes of the graph.

e) What is the range of this function?

f) After how many years will the copier be worth only $10,000?

4. A graph of a function, f, is shown. Find. Note the scales on the axes.

5. Determine whether the equation represents y as a function of x.

a. b.

Section 4.1-4.2 Solving Systems of Equations

Tutorials:

1. Which ordered pair is the solution to the given system of equations?

A. B. C. D.
2. Solve the system of equations given by and .
A. no solution B. C. D. infinitely many solutions

3. A store sells sheets for either $15 or $30. One month, sales totaled $12,570. If the store sold 563 sheets, how many $15 sheets were sold?

4. The difference of the measures of two complementary angles is 6 degrees. Find the measure of each angle. (Complementary angles add up to 90 degrees).

5. Solve the following systems:

a. b.

c. d.

Section 4.1-4.2 Solving Systems of Equations (continued)

6. The graph below gives the solution to which system of equations? You may assume that each tick mark represents one unit.

A. B. C. D.

7. What is true of the graph below? There may be more than one answer.

A. The graph represents a system of equations that is inconsistent.

B. The graph represents a system of equations with infinitely many solutions.

C. The graph represents a system of equations with no solution.

D. Both of these lines have positive slope.

Section 4.1-4.2 Solving Systems of Equations (continued)

8. Find the graph of the system. Use the graph to determine if the system is consistent or inconsistent. If the system is consistent, determine the number of solutions.

A. B.

Consistent, infinitely many solutions Consistent, one solution

C. D.

Consistent, infinitely many solutions Inconsistent

9. A group of 51 people attend a ball game. There were twice as many children as adults in the group. Set up a system of equations that represents the numbers of adults and children that attended the game and solve the system to find the number of children that were in the group.

A. B. C. D.

Section 5.3 Polynomials and Polynomial Functions

1. Evaluate each polynomial for the given value of the variable:

2. Add or subtract as indicated:

a) Subtract from

b) Add and.

3. Which expression is not a polynomial?

A. B. C. D.

4. What is the degree of 3a – 4a2b3 + b3 – 4ab?

A. 1B. 2C. 3D. 5

5. Write a polynomial to represent the area of a square with a side x minus the area of a triangle with a base 2x and a height of 5.
A. 4x B. C. D.

Section 5.3 Polynomials and Polynomial Functions

(continued)

6. Anthony and Sanford each throw a football. The height of Anthony's throw can represented by the equation A = –10x2 + 15x + 22, where A is height and x is the time in seconds. The height of Sanford's throw can represented by the equation

S = –9x2 + 14x + 23. At time x, how much higher is Sanford's throw?

A. x2 + x + 1 B. x2 – x – 1

C. x2 + x – 1 D. x2 – x + 1

7. Suppose the perimeter of a triangle is given by P = 13x + 5y, and two of its sides have lengths of 2x + y and 8x + 4y. What is the length of the third side?
A. 23x+5y B. 6x + 3y C. 3x D. 3x + y

8. The calculated annual fixed cost (in dollars) of owning and operating an automobile is given by the polynomial, where x is the number of years after 1999. The variable costs are represented by the polynomial. Find a polynomial that represents the difference between the fixed cost and the variable cost for a given year.

9. Multiply:

a. =

Section 5.4 Multiplying Binomials

1. Multiply as indicated:

2. For (a) and (b) below, find, , and :

Multiplying binomials:

Section 5.5-5.7 Factoring Polynomials

1. Factor out the greatest common factor:

2. In a polygon with n sides, the interior angles, measured in degrees, add up to

180n -360. Find an equivalent expression by factoring out the GCF.

3. The area (in square meters) of a pool is given by the expression , where l is the length of the pool.

a) Factor the expression for the area.

b) The width of this pool is 20 m. What is the length?

4. Factor:

Section 5.5-5.7 Factoring Polynomials (continued)

5. Find the missing factor:

6. Factor, if possible:

7. Factor, if possible:

Summary of factoring polynomials:

Section 5.5-5.7 Factoring Polynomials (continued)

True or False:

1. is a prime polynomial.

2. may be factored as a difference of two squares.

3. is a perfect square trinomial.

4. .

5. is a prime polynomial.

6. .

Choose the correct answer:

8. Factor 8a4b - 3a2b3c.
a. 8ab(a2 - 3bc) b. abc(8a3 - 3b) c. a2b(8a2 - 3bc) d. ab(8a3 - 3b2c)

9. Factor 2r3 + 8r2 - 10r completely.

a. 2(r3 + 4r2 - 5r) b. (2r - 1)(r + 5) c. 2r(r - 1)(r + 5) d. 2r(r2 + 4r - 5)

10. Factor m3 - 4m2 + 6m - 24.

a. m(m2 - 4m + 6) – 24
b. m2(m - 4) + 6m – 24
c. (m - 4)(m2 + 6)
d. (m + 4)(m2 - 6)

Section 5.8 Solving Equations by Factoring

1. Solve each equation:

2. Find all values such that:

3. A ball is dropped from a balloon 900 feet above the ground. The height, h, of the ball above the ground (in feet) after t seconds is given by the equation. When will the ball hit the ground?

Self-quiz on solving quadratic equations:

Factoring Jepoardy:

Section 6.1 Properties of Rational Expressions and Functions

1. Identify the value(s) for which the given expression is undefined:

2. Evaluate the expression for the given values of x: .

3. Reduce to lowest terms:

4. Fill in the expression that makes the rational expressions equivalent:

5. Fill in the expression that makes the rational expressions equivalent:

6. Find the domain in interval notation:

Section 6.2 Multiplication and Division of Rational Expressions

1. Perform the indicated operation and express the answers in lowest terms:

2, For , find :

Section 6.2 Multiplication and Division of Rational Expressions

(continued)

7. Write an expression for the area and simplify:

Section 6.3Addition and Subtraction of Rational Expressions

1. Find the least common denominator of .

2. Find the least common denominator of .

3. Write each expression using the LCD as the denominator: .

4. Write each expression using the LCD as the denominator: .

5. Perform the indicated operation and simplify where possible:

Section 6.3 Addition and Subtraction of Rational Expressions

(continued)

6. Given :

7. The formula is used by optometrists to determine how strong to make eyeglasses. If a =10 and b =0.2, find the value of P.

8. Write an expression for the sum of the reciprocals of two consecutive integers and simplify it.

9. Write an expression that represents the perimeter of the figure and simplify.

10. If represents the probability of an event, then the probability that the event will not occur is . Write as a single rational expression.

Some tutorials:

Finding LCD (whole numbers)

Adding Rational Expressions

Adding/Subtracting Rational Expressions

Section 6.5 Division of Polynomials

1. Divide as indicated:

2. Find:

3. To find the average cost of producing an item, divide the total cost by the number of items produced. A company that manufactures computer disks uses the function C(x)=200+2x to represent the cost of producing x disks.

a) Find the average cost.

b) What happens to the average cost as more items are produced? Hint: Set up a table.

4. Given ,

a) Evaluate

b) Divide .

c) Compare the value found in (a) to the remainder found in (b). What do you notice?

d) Evaluate if and then divide P(x) by. Can you state a rule that seems to work? This is called the Remainder Theorem.

Section 6.6-6.7 Solving Equations Involving Rational Expressions

and Applications

1. Solve: .

2. Solve: .

3. Solve: .

4. Find all values of x that satisfy and .

5. The function models the average cost per unit, f(x), for Electrostuff to manufacture x units of Electrogadget IV. How many units must the company produce to have an average cost per unit of $400? (Round to the nearest unit.)

6. At a warehouse, it takes two employees hours to load a truck when they work together. If it takes one employee working alone 3 hours to load a truck, how long does it take the other employee working alone?

Section 6.6-6.7 Solving Equations Involving Rational Expressions

and Applications (continued)

7. Solve the following equations:

8. If , find all values for which f(x)=3.

9. a. Solve for .

b. Solve for y: .

10. If , find all values for which f(t)=g(t).

11. The shadow cast by a yardstick is 2 ft. long. The shadow cast by a tree is 11 ft. long. Find the height of the tree.

12. One construction crew can pave a road in 24 hours, and a second crew can do the same job in 18 hours. How long would it take the two crews working together?

Sec. 7.1 Radicals

Some tutorials:

1. Simplify: .

2. Simplify: .

3. Simplify:

4. Simplify: .

5. Simplify:

6. True or false: .

7. True or false: .

8. True or false: .

9. What is the domain of ? State the answer in interval notation.

10. What is the domain of ? State the answer in interval notation

Sec. 7.1 Radicals (continued)

11. Write each of the expressions in simplified form:

a. e.

b. f.

c. g.

d. h.

12. Find the value of each root:

Sec. 7.2 Rational Exponents

1. Simplify each expression:

2. Simplify a) b)

3. Write as a single radical: .

4. The radius of a sphere of volume, V, is given by . Find the radius of a sphere having a volume of 85 in.3 (Round to the nearest 0.1 in.)

5. The geometric mean is a statistic used in business and economics. The geometric mean of three numbers is given by the expression, where p is the product of the three numbers. Find the geometric mean of 9, 3, and 8.

Sec. 7.3 Adding, Subtracting, and Multiplying Radicals

Combine the following expressions, if possible. Assume that any variables under an even root are nonnegative.

a. f.

b. g.

c. h.

d. i.

e. j.

2. Find:

3. The ramp shown at the right has a base, b, of 8 feet and a height, h, of 4 feet. The length of the ramp is given by. Find the length of the ramp.

4. True or false:

Sec. 7.3 Adding, Subtracting, and Multiplying Radicals

(continued)

5. Multiply or divide as indicated and simplify. Assume that any variables under an even root are nonnegative.

6. Find the areas of the figures below:

a. b.

Sec. 7.4 Quotients, Powers, and Rationalizing Denominators

1. Rationalize the denominator: .

A. B. C. D.

2. Rationalize the denominator: .

A. B. C. D.

3. Rationalize the denominator: .

A. B. C. D.

4. Simplify: .

A. B. C. D.

5. Rationalize the denominator and simplify: .

A. B. C. D.

6. Given the area of a circle, the formula can be used to calculate the radius of the circle. Rewrite the formula.

A. B. C. D.

7. Simplify:

Sec. 7.5 Solving Equations with Radicals and Exponents

1. Solve: .

A) B) C) D)

2. Solve: .

3. Solve the following problem:

4. Solve: .

A) B) C) D)

5. Solve for :

6. Solve and check each equation.

a. b. c.

7. Solve and check each equation.

a. b. c.

8. An airline’s cost for x thousand passengers to travel round trip from New York to Atlanta is given by , where is measured in millions of dollars and . Find the airline’s cost for 10,000 passengers. If the airline charges $320 per passenger, find the profit made by the airline for flying 10,000 passengers from New York to Atlanta.

Sec. 7.6 Complex Numbers

1. Simplify the following as much as possible:

2. Perform the addition or subtraction and write the result in standard form.

a. b.

3. Perform the operation and write the result in standard form.

a. b. c.

4. Write the quotient in standard form.

a. b. c.

5. Solve:

a. b. c.

6. True or false:

a. Every real number is also a complex number.

b. Every complex number is also a real number.

f. The complex conjugate of is .

g. The complex conjugate of is .

7. Show that is a solution to the equation .

Sec. 8.1 Factoring and Completing the Square

Tutorials:

1. Solve:

A. B. C. 324D.

2. Solve: .

A. B. C. D.