College Textbook Sample 4 by Terri Wright

Careers and Mathematics PSYCHOLOGIST Many investors, whether they are individuals with a few hundred dollars to invest or large institutions with millions, use securities and financial sales agents when buying or selling stocks, bonds, shares in mutual funds, annuities, or other financial products. Securities and financial services sales agents held about 281,000 jobs in 2004. The overwhelming majority of workers in this occupation are college graduates, with courses in business administration, gists is e for all : ycholo ag economics, utlook t of ps an the aver n e Job O m y h emplo w faster t . mathematics, and ll a r e gro ling, Ov 14 ounse gh 20 ted to ical, c $54,950. expec tions throu n finance. li c f so ere pa occu rning in 2004 w he field s t ual ea n ann ychologist orking in . ia d e s w 0 M hool p alf of those and $71,88 c s d n a h 0 iddle en $41,85 m e h e T d betw earne

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Graphing, Writing Equations of Lines; Functions; Variation

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3.1 The Rectangular Coordinate System 3.2 Graphing Linear Equations 3.3 Slope of a Nonvertical Line 3.4 Point-Slope Form 3.5 Slope-Intercept Form 3.6 Functions 3.7 Variation ■

Projects CHAPTER REVIEW CHAPTER TEST CUMULATIVE REVIEW EXERCISES

 In this chapter In this chapter, we will discuss equations with two variables. We will see that the relationships set up by these equations can also be expressed in tables or graphs.

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3.1 The Rectangular Coordinate System

Reach for Success Reading this Mathematics Text-

24-hour pie chart

______ working at outside employment? (include commute time) ______ sleeping? ______ preparing meals, eating and exercising? ______ getting ready for work/class?

Vocabulary

How can your schedule your time to allow for a successful semester? Begin by completing the pie chart to analyze a typical day. How can your schedule your time to allow for a successful semester? Begin by completing the pie chart to analyze a typical day.

Getting Ready

We can all agree that we only have 24 hours in a day, right? The expression “My, how time flies!” is certainly relevant. It might surprise you where the time goes when you begin to count the hours.

HOW MANY HOURS A DAY DO YOU SPEND: There are no right or wrong answers. You just need an honest assessment of your time. For every hour of activity, shade that many pie segments. (It might be helpful to use a different color to shade for each question.

3.1

The Rectangular Coordinate System and Graphing Pairs Objectives

IF YOU'RE "IN IT TO WIN IT", LET'S TRY THIS:

Section

1 Graph ordered pairs and mathematical relationships. 2 Interpret the meaning of graphed data 3 Interpret information from a step graph.4 In this section, the following words will be introduced. Be sure to learn the meaning of each word: rectangular coordinate system Cartesian coordinate system perpendicular lines x-axis y-axis

coordinates ordered pairs exponents directional

origin coordinate plane Cartesian plane quadrants x-coordinate y-coordinate

Graph each set of numbers on the number line. 1. –2, 1, 3

3. All numbers less than or equal to 3

negative

positive

2. All numbers less than or equal to 3

4. All numbers between -3 and 2

less than

greater than

______ in class? (include lab hours or tutoring time) ______ with family and friends? ______ on the internet, phone, playing video games, texting, watching tv, going to movies, or other entertainment

This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph.

1 Graph ordered pairs and mathematical relationships Now we have a mathematics problem! How many hours are left for studying? Remember, the rule-ofthumb is for every one hour in class, you will need three hours of studying. For mathematics, it could take even longer. The key is to spend as much time as you need to understand the material. There is no magic formula!

When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. • An equation can be true: 6 + 3 = 9. • An equation can be neither true nor false. For example, x + 5 = 15 is neither

true nor false because we don’t know what number x represents.

A Successful Study Strategy . . .

Adjust your schedule as needed to support your educational goals. At the end of the chapter you will find an additional exercise to help guide you to planning for a successful semester.

A rectangular coordinate system (see Figure 3-2) is formed by two perpendicular number lines. Recall that perpendicular lines are lines that meet at a angle. Recall that perpendicular lines are lines that meet at a angle. Recall that perpendicular lines are lines that meet at a angle. • The horizontal number line is called the x-axis. • The vertical number line is called the y-axis.

The positive direction on the x-axis is to the right, and the positive direction on the y-axis is upward. The scale on each axis should fit the data. For example, the axes of the graph designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph.

8 CHAPTER 3 Graphing, Writing, Equations of Lines; Functions; Variation

3.1 The Rectangular Coordinate System

Interpret the meaning of graphed data

Teaching Tips

If your class is full and the chairs are arranged in rows and columns, introduce graphing this way: 1. Number the rows and columns. 2. Ask row 3, column 1 to stand. This is the point (3, 1). 3. Ask (1, 3) to stand. That this is a different point emphasizes the ordered pair idea. 4. Repeat with more examples.

1. An equation can be true: 6 + 3 = 9. 2. An equation can be neither true nor false. For example, x + 5 = 15 is neither true nor false because we don’t know what number x represents. A rectangular coordinate system (see Figure 3-2) is formed by two perpendicular number lines. Recall that perpendicular lines are lines that meet at a angle. Recall that perpendicular lines are lines that meet at a angle. Recall that perpendicular lines are lines that meet at a angle.

The positive direction on the x-axis is to the right, and the positive direction on the y-axis is upward. The scale on each axis should fit the data. The point where the axes cross is called the origin. The two axes form a coordinate plane (often referred to as the Cartesian Plane) and divide it into four regions called quadrants, shown in Figure 3-2.

Ordered Pair Coordinates The first number in the pair is the x-coordinate, and the second number is the y-coordinate. Some examples of ordered pairs are (3, –4), , and (0, 2.5).

• The horizontal number line is called the x-axis. • The vertical number line is called the y-axis.

INTERCEPTS OF A LINE

The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y.

(a)

(b) Figure 3.1

EXAMPLE 1 An equation is a statement indicating that two expressions are equal. An example is

The process of locating a point in the coordinate plane is called graphing or plotting the point. In Figure 3-3(a), we show how to graph the point A with coordinates of (3, –4). Since the y-coordinate is negative, we then move down 4 units to locate point A.

x + 5 = 15. The equal symbol separates the equation into two parts.

on the x-axis, and move 3 units up. Point A lies in quadrant II. (See Figure 3-4.) b. To plot point B with coordinates of , we start at the origin and move 1 unit to the left and units down. Point B lies in quadrant III, as shown in Figure 3-4.

Step 1: To graph point C with coordinates of (0, 2.5), we start at the origin and move 0 units on the x-axis and 2.5 units up. Point C lies on the y-axis, in Figure 3-4. Step 2: To graph point D with coordinates of (4, 2), we start at the origin and move 4 units to the right and 2 units up. Point D lies in quadrant I, as shown in Figure 3-4. Analyze the problem

An equation is a statement indicating that two expressions are equal. An example is x + 5 = 15. The equal symbol separates the equation into two parts: The sides of an equation can be reversed, so we can write x + 5 = 15 or 15 = x + 5. TABLE 1 Equations C

Form an equation

–6

= 2•B1 – 3•C1 + 4

–1

= A2/(B2 – C2)

Subtract half the coefficient of x and add it to both sides. Factor and combine like terms.

An equation is a statement indicating that two expressions are equal. An example is x + 5 = 15. The equal symbol separates the equation into two parts.

Check + 5 = 15 or 15 = x + 5.

a SELF CHECK

Plot the points a. E(2, –2) 4 b. F(–4, 0) 0 c.

6 d. H(0, 5) 9

Solution a. To plot point A with nates (–2, 3), we start at the origin, move 2 units to the left

In an ordered pair, the x-coordinate is listed first.

Reneé Descartes (1596 - 1650)

Descartes is famous for his work in mathematics. His philosophy is expressed in the words, “I think, therefore I am.” He is best known in mathematics for his invention of a coordinate system and his work with conic sections.

(3, –4) The y-coordinate is listed second.

Point A is the graph of (3, –4) and lies in quadrant IV. The process of locating a point in the coordinate plane is called graphing or plotting the point. The process of locating a point in the coordinate plane is called graphing or plotting the point. In Figure 3-3(a), we show how to graph the point A with coordinates of (3, –4).

Terminating Decimals 1 — = 0.5 2

Repeating Decimals 1 — = 0.33333 . . . or 0.3 3

3 1 — = 0.75 — = 0.66666 . . . or 0.16 4 6

Comment Note that point A with coordinates of (3, –4) is not the same as point B with coordinates (–4, 3). Since the order of the coordinates of a point is important, we call the pairs ordered pairs. Point A is the graph of (3, –4) and lies in quadrant IV. The process of locating a point in the coordinate plane is called graphing or plotting the point. The point where the axes cross is called the origin. This is the 0 point on each axis. The two axes form a coordinate plane (often referred to as the Cartesian Plane) and divide it into four regions called quadrants, which are numbered as shown in Figure 3-2.

10 CHAPTER 3 Graphing, Writing, Equations of Lines; Functions; Variation The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y.

a. To plot point A with nates (–2, 3), we start at the origin, move 2 units to the left on the x-axis, and move 3 units up. Point A lies in quadrant II. (See Figure 3-4.) b. To plot point B with coordinates of , we start at the origin and move 1 unit to the left and units down. Point B lies in quadrant III, as shown in Figure 3-4. Step 1: To graph point C with coordinates of (0, 2.5), we start at the origin and move 0 units on the x-axis and 2.5 units up. Point C lies on the y-axis, as shown in Figure 3-4. Step 2: To graph point D with coordinates of (4, 2), we start at the origin and move 4 units to the right and 2 units up. Point D lies in quadrant I, as shown in Figure 3-4.

Analyze the problem

An equation is a statement indicating that two expressions are equal. An example is x + 5 = 15. The equal symbol separates the equation into two parts: The expression x + 5 is the left side and 15 is the right side. The letter x is the variable (or the unknown). The sides of an equation can be reversed, so we can write x + 5 = 15 or 15 = x + 5. Equations C

Form an equation

a SELF CHECK FINDING SIGNIFICANT DIGITS

–6

= 2•B1 – 3•C1 + 4

–1

= A2/(B2 – C2)

Subtract half the coefficient of x and add it to both sides.

3.2

Graphing Linear Equations and Interpreting Information on a Graph 1 Determine whether an ordered pair satisfies an equation in two

Factor and combine like terms.

An equation is a statement indicating that two expressions are equal. An example is x + 5 = 15. The equal symbol separates the equation into two parts: The expression x + 5 is the left side and 15 is the right side. The letter x is the variable (or the unknown). The sides of an equation can be reversed, so we can write x + 5 = 15 or 15 = x + 5.

Plot the points a. E(2, –2) 4 b. F(–4, 0) 0 c.

6 d. H(0, 5) 9

The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y.

Section

Objectives

Solution

GRAPHING POINTS Plot the graphing points a. A (–2, 3) b. A (–2, 3 c. C (0, 2.5) d. D (4, 2).

Vocabulary

EXAMPLE 1

The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. 1. To plot point A with nates (–2, 3), start at the origin, move 2 units to the left. 2. To plot point B with coordinates of , we start at the origin and move 1 unit . 3. To graph point C with coordinates of (0, 2.5), we start at the origin and move 0 units on the x-axis and 2.5 units up. The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y.

RULES FOR THE ORDER OF OPERATIONS

2 3 4 5 6

variables Use an equation to complete a table of values Graph a linear equation in two variables using a table of values Graph a linear equation in two variables using the intercept method Graph a horizontal line and a vertical line Write a linear equation in two variables from real-world information and then use the graph to answer questions

In this section, the following words will be introduced. Be sure to learn the meaning of each word: input value output value dependent variable,

independent variable x-intercept y-intercept

Getting Ready

INTERCEPTS OF A LINE

3.2 Graphing Linear Equations

Graph each set of numbers on the number line.

Determine Whether Ordered Pairs Satisfy an Equation

1. –2, 1, 3 3 = y–2

2. Numbers less than or between ______ to 3

less than or equal to 3 3. All numbers ________

4. All numbers ________ -3 and 2

equal

5. Find five pairs of numbers with a sum of 8 3 = y–2 6. Find five pairs of numbers with a difference of 5 3 = y–2

Interpret the Meaning of Graphed Data When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is sometimes called a Cartesian coordinate system after the 17th-century French mathematician René Descartes. When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is sometimes called a Cartesian coordinate system after the 17th-century French mathematician René Descartes.

When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is sometimes called a Cartesian coordinate system after the 17th-century French mathematician René Descartes. his graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is sometimes called a Cartesian coordinate system after René Descartes.

12 CHAPTER 3 Graphing, Writing, Equations of Lines; Functions; Variation

(1)

y=5-x x y (x, y) 1

(1,4) (1,8) (1,8)

The scale on each axis should fit the data. For example, the axes of the graph of the arch shown in Figure 3-1(b) are scaled in units of 100 feet.

The cost of good hammers

the cost of plus equals better hammers

the cost of best hammers

+ = 6y

520

Rectangular coordinate system (see Figure 3-2) is formed by two perpendicular number lines. Recall that perpendicular lines are lines that meet at a angle.

Accent

The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. 1. To plot point A with (–2, 3), we start at the origin, move 2 units to the left. 2. To plot point B with coordinates of , we start at the origin and move 1 unit . 3. To graph point C with coordinates of (0, 2.5), we start at the origin and move 0 units on the x-axis and 2.5 units up.

on technology  Generating Tables of Solutions

To use a graphing calculator to evaluate the determinaant in Example 3, we first enter the matrix by pressing the MATRIX key, selecting EDIT, and pressing the ENTER key. To use a graphing calculator to evaluate the determinaant in Example 3, we first enter the matrix by pressing the MATRIX key, selecting EDIT, and pressing the ENTER key.

Proof

Comment Note that point A with coordinates of (3, –4) is not the same as point B with coordinates (–4, 3).

An equation is a statement indicating that two expressions are equal. An example is x + 5 = 15. The equal symbol separates the equation into two parts: The expression x + 5 is the left side and 15 is the right side. An equation is a statement indicating that two expressions are equal. The letter x is the variable (or the unknown). The expression x + 5 is the left side and 15 is the right side. The letter x is the variable (or the unknown).The sides of an equation can be reversed, so we can write x + 5 = 15 or 15 = x + 5. The expression x + 5 is the left side and 15 is the right side. The letter x is the variable (or the unknown).The sides of an equation can be reversed, so we can write x + 5 = 15 or 15 = x + 5.

Lewis Carroll When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular system. This coordinate system is sometimes called a Cartesian coordinate system after the 17th-century French mathematician René Descartes. When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate Alice confronts the Queen of Hearts system. The graph, who threw the deck. shown in Figure 3-1(b), is drawn on a grid. When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid. This coordinate system is sometimes coordinate system is sometimes called © Jean Kiery/Corbis

Everyday connections

Section

3.3

Graph each set of numbers on the number line. 1. –2, 1, 3

negative

2. All numbers less than or equal to 3

less than

Source: http://mathworld.wolfram.com/Determinant. html

a Cartesian coordinate system after the 17th-century French mathematician René Descartes. This coordinate system is sometimes coordinate system is sometimes called a Cartesian coordinate system after the 17th-century mathematician René Descartes. When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is sometimes called a Cartesian coordinate system after the 17th-century mathematician René Descartes. When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is called a Cartesian coordinate system after the 17th-century mathematician René Descartes.

The Rectangular Coordinate System and Graphing Pairs Objectives

Determine whether an ordered pair satisfies an equation in two variables

Vocabulary

3.3 The Rectangular Coordinate System and Graphing Pairs

origin coordinate plane Cartesian plane quadrants x-coordinate y-coordinate

coordinates ordered pairs exponents directional

Getting Ready

14 CHAPTER 3 Graphing, Writing, Equations of Lines; Functions; Variation

3.6 Exercises

Graph each set of numbers on the number line. 1. –2, 1, 3

3. All numbers less than or equal to 3

3.6 Exercises `

WARM UPS Solve each equation.

negative

positive

2. All numbers less than or equal to 3

4. All numbers between -3 and 2

less than

greater than

Write a linear equation in two variables from real-world information and then use the graph to answer questions When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is sometimes called a Cartesian coordinate system after the 17th-century French mathematician René Descartes. • The horizontal number line is called the x-axis. • The vertical number line is called the y-axis.

Perspective

When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model.

SUBHEAD A

This graph, shown in Figure 3-1(b), is drawn on a grid called the rectangular coordinate system. This coordinate system is sometimes called a Cartesian coordinate

a SELF CHECK ANSWERS

To the Instructor 1. Problem 1 reinforces the concept of (0,0) as the y-intercept 2. Problem 2 requires the original equation to be written in slope-intercept form to finding the needed slope 3. This problem requires the student to distinquish between independent and independent variables and then develop their own reasonable scale to plot the data.

system after the 17th-century French mathematician René Descartes. When designing the Gateway Arch, shown in Figure 3-1(a), architects created a mathematical model of the arch called a graph. This coordinate system is sometimes called a Cartesian coordinate. Source: http://mathworld.wolfram.com/Determinant. html

1. a. y = -2x+7 b. 6 ft, c. 4 days 2. no 3. (-10) 4. (-10, 12)=8x 5. no 6. (-10, 12) 7. (-10, 12)=8x 9. a. y = -2x+7 b. 6 ft, c. 4 days 10. no 11. (-10, 12)

NOW TRY THIS 1. Find the slope and y-intercept of the line y = x, m = 1; (0,0) slope = 3y -20 2. Write an equation of the line passing through (2, –6) and perpendicular

to 2x+3y = 12. y = x, m = 1; (0,0) 3. Set up a coordinate plane with an appropriate scale and plot the points in the exercise

facts described below. Damon paid $56 for 2 tickets to a sports event. Javier paid the 88 tickets for 3 tickets to the same event. Caroline paid $124 for 4 tickets courtside.

1. 5x + 4 = 14 3. 5x + 4 = 14 5. 5x + 4 = 14 7. 5x + 4 = 14

< 28-10 < 28-10 < 28-10 < 28-10

2. 5x + 4 = 14 4. 5x + 4 = 14 6. 5x + 4 = 14 8. 5x + 4 = 14

x > 32 x > 32 x > 32 x > 32

REVIEW Perform the operations. Simplify the results. 9. a. Adding the same number to both sides of an inequality does not change the solutions. b. Multiplying or dividing both sides of an inequality by the same positive number does not change the solutions. c. If we multiply or divide both sides of an inequality by a negative number, the direction. 10. To solve properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side. x > 32. 11. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side. x > 32 12. The solution set of an inequality is graphed below. Which of the four numbers, 3, –3, 2, and 4.5, when substituted for the variable in that inequality, would make it true? C

–6

= 2•B1 – 3•C1 + 4

–1

= A2/(B2 – C2)

26. 5x + 4 = 14 expression 27. 5x + 4 = 14 expression 28. 5x + 4 = 14 equation 29. 5x + 4 = 14 expression 30. Write each symbol. a. 5x + 4 = 14 < 28-10 b. 5x + 4 = 14 c. 5x + 4 = 14 d. 5x + 4 = 14 < 28-10 31. Consider the graph of the interval [4, 8). a. Is the endpoint included or not included in the graph? included b. Is the endpoint 8 included or not included in the graph? not included Use a graphing calculator to solve the addition or subtraction property of equality for each equation. Check all solutions. SEE EXAMPLES 3-4 (OBJECTIVE 3) 32. Adding the same number to both sides of an inequality does not change the solutions.

Source: New York Times, December 24, 2001, p. A4.

Complete the solution to solve each inequality. 33. Adding the same number to both sides of an inequality does not change the solutions. Source: New York Times, December 24, 2001, p. A4.

ADDITIONAL PRACTICE Solve each equation.

13. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side. x > 32 14. The solution set of an inequality is graphed below. Which of the four numbers, 3, –3, 2, and 4.5, when substituted for the variable in that inequality, would make it true? The solution set of an inequality is graphed below.

VOCABULARY & CONCEPTS Fill in the blanks 15. Consider the graph of the interval [4, 8). a. Is the endpoint included or not included in the graph? included b. Is the endpoint 8 included or not included in the graph? not included 16. Adding the same number to both sides of an inequality does not change the solutions. 17. Adding the same number to both sides of an inequality does not change the solutions.

GUIDED PRACTICE Determine whether each statement is an expression or an equation. SEE EXAMPLE 1 (OBJECTIVE 1) 18. 5x + 4 = 14 equation 20. 5x + 4 = 14 expression 22. 5x + 4 = 14 equation 24. 5x + 4 = 14 equation

19. 5x + 4 = 14 equation 21. 5x + 4 = 14 expression 23. 5x + 4 = 14 expression 25. 5x + 4 = 14 equation

34. Write each symbol. a. 5x + 4 = 14 inequality b. 5x + 4 = 14 c. 5x + 4 = 14 d. 5x + 4 = 14 < 28-10 35. Consider the graph of the interval [4, 8). a. Is the endpoint included or not included in the graph? included b. Is the endpoint 8 included or not included in the graph? not included

APPLICATIONS Solve each application problem involving percents. SEE EXAMPLES 13-14 (OBJECTIVE 7) . 36. Selling microwave ovens To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side. x > 32.

SOMETHING TO THINK ABOUT Solve each equation. 37. Hospitals Rewrite the inequality 32 < x in an equivalent form with the variable on the left side. The solution set of an inequality is graphed below. Which of the four numbers, 3, –3, 2, and 4.5

16 CHAPTER 3 Graphing, Writing, Equations of Lines; Functions; Variation

Projects

three digits in either its numerator or its denominator) that is closer to . Who in your class has done best? Realizing that $5 could not be evenly divided among the three professors.

PROJECT 1 The circumference of any circle and its diameter are related. When you divide the circumference by the diameter, the quotient is always the same number, pi, denoted by the Greek letter. ■■

■■

arefully measure the circumference of several C circles—a quarter, a dinner plate, a bicycle tire— whatever you can find that is round. Use the key on the calculator to obtain a more accurate value of . How close were your approximations?

PROJECT 2 a. The fraction is used as an approximation of. To how many decimal places is this accurate? b. Experiment with your calculator and try to do better. Find another fraction (with no more than

Summary

3 Chapter Test

Write an essay answering this question.

Equivalent equations have the same solutions.

When three professors attending a convention in Las Vegas registered at the hotel, they were told that the room rate was $120. Each professor paid his $40 share. Each professor paid his $40 share. Later the desk clerk realized that the cost of the room should have been $115. Later the desk clerk realized that the cost of the room should have been $115. To fix the mistake, she sent a bellhop to the room to refund the $5 overcharge. Realizing that $5 could not be evenly divided among the three professors. Since each professor received a $1 refund, each paid $39 for the room, and the bellhop kept $2. This gives , or $119. What happened to the other $1?

SECTION 2.1 Solving Equations Using Properties of Equality 3y – 1 = 2y + 7

Solving for the Variable: A number that makes an equation a true statement when substituted for the variable is called a solution of the equation.

Determine if 2 is a solution of xx – y. Check: Substitute 2 for each x. 6= 6 Since the resulting statement is true, 2 is a solution

An equation statement indicating that two expressions are equal. The equal symbol = separates an equation into two parts: the left side and the right side

True.

3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7

REVIEW EXERCISES Determine whether the number is a solution of the equation.

1. 5x + 4 = 14 2 2. 5(2x – 4) – 5x = 0 4 3. 5(2x – 4) – 5x = 0 4 4. 5x + 4 = 14 5. 0.5 – 0.02(y – 2) = 0.16 + 0.36y 5 6. 3(a + 8) = 6(a + 4) – 3a 7. 2(y + 10) + y = 3(y + 8) 4

8. An inequality is a statement containing the symbols: >, of x > 2 can be expressed in interval notation as (2, ∞). 9. To solve an inequality means to find all the values of the variable that make the inequality true. 10. The solution set of x > 2 can be expressed in interval . 11. The inequality is an example of a compound inequality.

SECTION 2.2 Solving Equations Using Properties of Equality DEFINITIONS AND CONCEPTS

EXAMPLES

n equation is a statement indicating that two A expressions are equal. The equal symbol = separates an equation into two parts: the left side and the right side

3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7

Substitute x +7 for y

27 – 1 18 + 7

Simplify

26 = 25

Distribute

A number that makes an equation a true statement when substituted for the variable is called a solution of the equation.

Determine if 2 is a solution of xx – y. Check: Substitute 2 for each x. 6= 6 True. Since the resulting statement is true, 2 is a solution

Equivalent equations have the same solutions.

3y – 1 = 2y + 7

EXAMPLES

eparators in an Equation: S An equation is a statement indicating that two expressions are equal. The equal symbol = separates an equation into two parts: the left side and the right side

Determining the Equation: Equivalent equations have the same solutions.

Determine if 2 is a solution of xx – y. Check: Substitute 2 for each x. 6= 6 Since the resulting statement is true, 2 is a solution

PROJECT 3

DEFINITIONS AND CONCEPTS

A number that makes an equation a true statement when substituted for the variable is called a solution of the equation.

3 (9) – 1 2(9) + 7

Substitute x +7 for y

27 – 1 18 + 7

Simplify

3 (9) – 1 2(9) + 7' 3 (9) – 1 2(9) + 7

26 = 25

Distribute

3 (9) – 1 2(9) + 7 Equations

–6

= 2•B1 – 3•C1 + 4

–1

= A2/(B2 – C2)

3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7 3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7

Substitute x +7 for y

27 – 1 18 + 7

Simplify

26 = 25

Distribute

" Test '

1. An inequality is a statement that contains one of the symbols: >, , <, or . 2. To solve an inequality means to find all the values of the variable that make the inequality true. 3. The solution set of x > 2 can be expressed in interval notation as (2, ∞). 4. The inequality is an example of a compound inequality. 5. a. Adding the same number to both sides of an inequality does not change the solutions.

b. Multiplying or dividing both sides of an inequality by the same positive number does not change the solutions. c. If we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed for the inequalities to have the same solutions.

3–4 Cumulative Review

Reach for Success EXTRA PRACTICE EXERCISE ANALYZING YOUR TIME Now that you’ve analyzed a day, let’s move on to take a look at a typical week. Fill in the chart below to account for every hour of every day. To simplify the process, you can use the following abbreviations: S (sleeping) W (work time including commute) C (class time including commute) F (time with family and friends) P (preparing for work; preparing meals, eating) ST (study time) and E (entertainment) Remember, there are no right or wrong answers. This information is to give you a complete picture of how you are spending your time in a typical week. SUNDAY

MONDAY

TUESDAY

WEDNESDAY

THURSDAY

FRIDAY

SATURDAY

6:00 – 7:00 7:00 – 8:00 8:00 – 9:00 9:00 – 10:00 10:00 – 11:00 11:00 - Noon Noon – 1:00 1:00 – 2:00 2:00 – 3:00 3:00 – 4:00 4:00 – 5:00 5:00 – 6:00 6:00 – 7:00 7:00 – 8:00 8:00 – 9:00 9:00 – 10:00 10:00 – 11:00 11:00 – 12:00 12:00 – 1:00 1:00 – 2:00 2:00 – 3:00 3:00 – 4:00 4:00 – 5:00 5:00 – 6:00

After reviewing your weekly schedule, is there enough time to meet the “rule of thumb” of studying 3 hours per week for every 1 hour you are in class? If yes, congratulations! If no, is it possible for you to find the additional hours to help you be successful in this and all your other classes? ______ Are you able/willing to adjust your schedule to find this time?_____ Explain your answer.

3–4

" Cumulative Review '

Determine whether the given number is a solution of the equation. 1. An inequality is a statement that contains one of the symbols: >, , <, or . 2. To solve an inequality means to find all the values of the variable that make the inequality true. 3. The solution set of x > 2 can be expressed in interval notation as (2, ∞). 4. The inequality is an example of a compound inequality. 5. a. Adding the same number to both sides of an inequality does not change the solutions. b. Multiplying or dividing both sides of an inequality by the same positive number does not change. c. If we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed for the inequalities to have the same solutions. 6. To solve , properties of inequality are applied to all three parts of the inequality. 7. Rewrite the inequality 32 < x in an equivalent form with

the variable on the left side. x > 32 8. The solution set of an inequality is graphed below. Which of the four numbers, 3, –3, 2, and 4.5, when substituted for the variable would make it true? 9. Write each symbol. a. < is less than or equal to b. infinity ∞ c. bracket [ or ] d. is greater than > 10. Consider the graph of the interval [4, 8). a. Is the endpoint included or not included in the graph? included b. Is the endpoint 8 included or not included in the graph? not included 11. Consider the graph of the interval [4, 8). a. Is the endpoint included or not included in the graph? included b. Is the endpoint 8 included or not included in the graph? not included 12. Determine whether each number a. yes b. –15 no