text stringlengths 235 3.08k |
|---|
top row contains ten bricks. How many bricks are there in the monument? 43. A ladder with nine rungs is to be built, with the bottom rung 24 inches wide and the top rung 18 inches wide. If the lengths of the rungs decrease uniformly from bottom to top, how long should each of the seven intermediate rungs be? second, 1... |
1b 6 As the positions within a sequence increase by one, the value of the terms increases by 5. Notice on the graph that as the x-values move to the right a distance of one, the y-values move up by a distance of 5. This common difference represents one of the most identifiable characteristics of a line, its slope. Slop... |
points with If the line through these points is the ratio y2 x2 change in y change in x ¢y ¢x x1 y1 x1 x2, then the slope of Example 1 Finding Slope Given Two Points Find the slope of the line that passes through ure 1.4-4. 1 0, 1 2 and (4, 1). See Fig- x Solution Apply the formula in the previous box with x2 4, y2 1.... |
ope-Intercept Form A nonvertical line intersects the y-axis at a point with coordinates (0, b), because every point on the y-axis has first coordinate 0. The number b is called the y-intercept of the line. For example, the line in Figure 1.4-4 has y-intercept because it crosses the y-axis at (0, 1). 1 y (x, y) y − b (0... |
line. The connection between the explicit form of an arithmetic sequence, a line, (n 1)˛d, is as follows. u1 un y mx b, and the slope-intercept form of • The slope of the line corresponds to the common difference of the sequence, m d. • The y-intercept represents the value of the first term of the sequence minus the d... |
the system is worthless, that is, the x-value that corresponds to a y-value of 0. Solution a. Linear depreciation means that the equation that gives the value y of the computer system in year x has the form for some constants m and b. Because the system is worth $7000 new (when x 0 ) the y-intercept is 7000 and the eq... |
a line, the point-slope form is easier to use, unless you know the y-intercept. 2. The point-slope form can also be used to graph a line because any point of the line can be used as the initial point and the remaining points can be found by using the equation’s slope. The slope determines how to find a second point fr... |
y 3 2 1 0 −1 1 3 4 Find the equation of the vertical line shown in Figure 1.4-12. −2 Solution Figure 1.4-12 Every point on the vertical line in Figure 1.4-12 has first coordinate 2. and the line is the graph Thus, every point on the line satisfies x 0y 2, 38 Chapter 1 Number Patterns of the equation x 2. (2, 1) and (2... |
equation is 2x 3y 1. y 2 3, Standard Form of a Line NOTE The standard form of a line is sometimes called the general form of a line and may also be written as Ax By C 0. The following box summarizes the different forms of the equation of a line and when each form is best used. Forms of Linear Equations The forms of th... |
18. m 2; 20. m 0; 2, 1 2 4, 5 1 1 2 In Exercises 21–24, find the equation of the line through the given points. 21. 0, 5 1 and 1 2 3, 2 2 22. 4, 3 2 1 and 2, 1 1 2 23. 4 3 a, 2 3b and 1 3 a, 3 b 24. (6, 7) and (6, 15) In Exercises 25–28, determine whether the line through P and Q is parallel or perpendicular to the li... |
the vertices 48. For a given arithmetic sequence, the common u1 difference is y-intercept of the graph of this sequence. Find the slope and 6. and 3 49. For a given arithmetic sequence, the common u1 difference is 8 and y-intercept of the graph of this sequence. Find the slope and 2. 50. Let L be a line that is neithe... |
x 0 poverty level income y in year x (with corresponding to 1981). b. Use the equation of part a to estimate the poverty level income in 1990 and 2005. 54. At sea level, water boils at 1100 ft, water boils at between boiling point and height is linear. a. Find an equation that gives the boiling point The relationship ... |
. Find an equation that gives the revenue r from selling x items. b. How many items must be sold for the company to break even (i.e., for revenue to equal cost)? variable costs of $8.50 per hat. a. Find an equation that gives the total cost y of producing x hats. b. What is the average cost per hat when 20,000 are made... |
relationship between two quantities. For instance, • Calculate finite differences How does money spent on advertising affect sales? and use residuals to determine the model of best fit • Use a calculator to determine a linear model • Find and interpret the correlation coefficient for a model • Create and interpret a r... |
.23 3.64 4.06 Determine whether a line would be a good model for this data. Use two different methods. a. Calculate the finite differences for the data points. b. Draw a scatter plot of the data. Solution Year Cash Flow Differences a. Subtract each cash flow from the preceding one and record the difference, as shown in... |
each other, the model is probably a reasonable one. However, this is not always enough to determine which of several models is best because their residuals may all have the same sum. Consequently, to find which model among several fits the data best, use the sum of the squares of the residuals because this sum has no ... |
1 1 2 or y 0.6 x 1.4 Figure 1.5-5 shows the graph of Model B and the table below shows its residuals. The sum of the squared residuals is 1.56. Data point (x, r) 0, 1 1, 2 2, 2 3, 3 4, 3 5, 5 6 Model point (x, y) Residual r y Squared residual (r y)2 1 1 1 1 1 0, 1.4 1, 2 2 1 2, 2.6 3, 3.2 4, 3.8 5, 4.4 6.4 0 0.6 0.2 0... |
Modeling Data A circle can be circumscribed around any regular polygon. The lengths of the radii of the circumscribed circles around regular polygons whose sides have length of one unit are given as follows. Number of sides Radius 3 4 5 6 7 8 9 0.577 0.707 0.851 1.00 1.152 1.306 1.462 a. Draw a scatterplot. b. Calcula... |
and x, r 1 1 2 1 2 Use a linear model when the scatter plot of the residuals shows no obvious pattern, as shown in Figure 1.5-8. Use a nonlinear model when the scatter plot of the residuals has a pattern, as shown in Figure 1.5-9. r y x x Figure 1.5-8 Figure 1.5-9 Technology Tip The regression equation is stored in a ... |
ups Sold” and “Total Revenue” in the chart. That is, the data points are (112, 202), (88, 119), and so on. Next, use the linear regression function on these lists to approximate the least–squares regression line. y 1.586x 0.895 Store its equation as 1.5-10a and 1.5-10b. y1 in the equation memory, as shown in Figure Fig... |
the cost equation found in part c on the same screen, and find the x-coordinate of their intersection (shown in Figure 1.5-13). Since 89.6 cups cannot be sold, 90 cups a day must be sold to break even. ■ 180 Figure 1.5-13 Example 5 Prediction from a Model The total number of farm workers (in millions) in selected year... |
line. So when r is negative, the regression line slants downward from left to right. In other words, as x increases, y decreases. In such cases, we say that the data has a negative correlation. When r is positive, the regression line slopes upward from left to right, and the data is said to have a positive correlation... |
insurance policy for a female nonsmoker. Let x represent age and y the amount of the premium. Age Premium 25 30 35 40 45 50 55 $11.57 $11.66 $11.83 $13.05 $16.18 $21.32 $29.58 10. The table shows the percent of persons in the United States below the U.S. poverty level in x 0 correspond to 1960. selected years. Let 54 ... |
which of the following equations models beef consumption and which one models poultry consumption. Confirm your answer by graphing. 717.46x 1,405,160 329.86x 632,699 y1 y2 15. The table at the bottom of the page gives the a. Find a linear model for this data, using x 0 to correspond to 1950. b. In the unlikely event t... |
860 $896 $924 Year 1996 1997 1998 1999 2000 2001 56 Chapter 1 Number Patterns 18. The table shows what percent of federal aid is given in the form of loans to students at a particular college in selected years. Year (in which school year begins) Loans (%) 1975 1978 1984 1987 1990 18 30 54 66 78 a. Find a linear model f... |
x) and consumption ( y) of primary energy in quadrillion BTUs for a sample of countries in 1995. Australia (7.29, 4.43) Brazil (4.55, 6.76) Canada (16.81, 11.72) China (35.49, 35.67) France (4.92, 9.43) Germany (5.42, 13.71) India (8.33, 10.50) Indonesia (6.65, 3.06) Iran (9.35, 3.90) Japan (3.98, 21.42) Mexico (8.15, ... |
42 501 603 539 515 513 542 599 539 499 488 527 a. Make a scatter plot of the percent of students who took the SAT (x) versus the average SAT math score (y). b. Find a linear model for the data. c. What is the slope of your linear model? What does this mean in the context of the problem? d. Below is the data on four add... |
previous term by which, 1 2 gives the following recursive function. u1 5 2 ˛ and un 1 2 ˛un1 for n 2 is a geometric sequence with common ratio r, then for each un6 If the term preceding 5 un is un1 and un un1 r, or equivalently, un run1. ■ n 2 Section 1.6 Geometric Sequences 59 Recursive Form of a Geometric Sequence I... |
4 is 1 2 Figure 1.6-2a The table in Figure 1.6-2b confirms the apparent equality of the two functions. ■ 7 2n1. un Figure 1.6-2b Explicit Form of a Geometric Sequence implies that The recursive formula for u2 u3 u4 u5 n 2, 3, 4, p u1r u2r u3r u4r u1r 2 u1r 2 u1r 3 1 1 1 r u1r2 r u1r3 2 r u1r 4 2 } is a geometric sequen... |
is just k times the constant. If a geometric sequence is not constant (that is, then its partial sums are given by the following formula. r 1, 2 Partial Sums of a Geometric Sequence The kth partial sum of the geometric sequence { mon ratio r 1 is un } with com- k a n1 un u1a 1 r k 1 r b Proof If S denotes the kth part... |
Sequences 63 Solution First consider how far the ball travels on each bounce. On the first bounce, it rises 6 feet and falls 6 feets for a total of 12 feet. On the second bounce it rises and falls of the previous height, i.e., it travels of 12 feet. The 2 3 2 3 distance traveled is a geometric sequence with u1 12 and ... |
, u2 6, r 1 4 In Exercises 19–22, show that the given sequence is geometric and find the common ratio. 19. n 1 2b ea 21. 5n2 5 6 f 20. 22. 23n 6 n 2 3 6 5 5 In Exercises 23–28, use the given information about and recursive the geometric sequence un6 5. and explicit formulas for un to find u5 23. u1 256, u2 64 24. u1 1 ... |
first day of 2¢ 40. Starting with your parents, how many ancestors do you have for the preceding ten generations? 41. A car that sold for $8000 depreciates in value 25% each year. What is it worth after five years? 42. A vacuum pump removes 60% of the air in a container at each stroke. What percentage of the original ... |
............. 6 Function................................... 7 Function notation........................... 9 Definition of a sequence..................... 13 Sequence notation.......................... 14 Recursively defined sequence................ 15 Recursive form of an arithmetic sequence...... 22 Explicit form of ... |
................... 39 Mathematical model........................ 43 Finite differences........................... 43 Residual.................................. 44 Least–squares regression line................. 47 Correlation coefficient....................... 47 Correlation and slope....................... 52 Section ... |
y y1 m x x12 1, x1, y12 1 is a given point on the line. Ax By C, where The standard form of the equation of a line is A, B, and C are integers. The equation of a vertical line has the form x h. The equation of a horizontal line has the form y k. Parallel lines have equal slopes. The product of the slopes of perpendicu... |
the function given by g r 2r 4 2r 2? 20. What is the domain of the function f x 1 2 2 1 2x 2? 21. The radius of an oil spill (in meters) is 50 times the square root of the time t (in hours). a. Write the rule of a function f that gives the radius of the spill at time t. b. Write the rule of a function g that gives the... |
. Roberta had $1525 in a savings account 2 years ago. What will be the value of her account 1 year from now, assuming that no deposits or withdrawals are made and the account earns 6.9% interest compounded annually? Find the solution using both a recursive and an explicit formula. 36. Suppose that $3,000 is invested at... |
c, and d such that 8, b, c, d, 23 are the first five terms of an arithmetic sequence. Section 1.4 49. The national unemployment rates for 1990–1996 were as follows. (Source: U.S. Department of Labor, Bureau of Labor Statistics) Year 1990 1991 1992 1993 1994 1995 1996 Rate (%) 5.6 6.8 7.5 6.9 6.1 5.6 5.4 Sketch a scatt... |
6. 61. The graph of 2y 8 3x has y-intercept 4. 62. The lines 3x 4y 12 and 4x 3y 12 are perpendicular. 63. Slope is not defined for horizontal lines. 64. The line in the figure at right has positive slope. 65. The line in the figure does not pass through Quadrant III. 66. The y-intercept of the line in the figure is ne... |
x 0 corresponding to 1950. b. Use the equation in part a to estimate the population of San Diego in 1975 and 2000. In Exercises 75 –78, match the given information with the graph, and determine the slope of each line. y y 300 200 100 1000 800 600 400 200 3 6 a. 9 12 x x y y 300 200 100 600 400 200 1 2 b. 3 4 x x 2 4 6... |
74 y3 0.21x 15.48 82. a. According to the models in Exercise 81, is the percentage of female or male managers increasing at the greater rate? 74 Chapter Review b. Use the models to predict the percentage of female managers and the percentage of male managers in the year 2000. c. What year do the models indicate that th... |
, d, 27 are the first four terms of a geometric sequence. 93. Is it better to be paid $5 per day for 100 days or to be paid the first day, 10 the second day, 20 the third day, and have your salary increase in this fashion every day for 100 days? 5¢ ¢ ¢ 94. Tuition at a university is now $3000 per year and will increase... |
of partial sums are getting closer and closer to 3. Consequently, 2 0.6 2 0.6 2 2 0.6 3 2 0.6 2 where 3 is said to be sum, or limit, of the infinite series, In the general case, an infinite series, or simply series, is defined to be an expression of the form a2 p an a5 a3 a4 p a1 in which each an is a real number. Thi... |
the 1, not convergent is said to be divergent. If 0 r 0 series is divergent. Therefore, a geometric series is only convergent 6 1. when 0 r 0 Determine whether the infinite geometric series converges. q n1 a. a n1 6 2 1 2 q b. a n1 8 n 5 77 Solution a. The first term is 6 and the common ratio is 2. The sum of the firs... |
3573 the number as p as an infinite series: 6.8 0.0573573573 p. Then consider 0.0573 0.0000573 0.0000000573 0.0000000000573 p, which is the same as 0.0573 0.001 1 21 0.0573 2 0.001 1 2 1 2 This is a convergent geometric series with sum is 0.0573 0.001 1 0.0573 2 3 1 and 0.0573 p 2 r 0.001. 2 a1 a1 1 r 0.0573 1 0.001 0.... |
be detonated at precisely the correct height at the right moment. The time needed for a rocket to reach a specific height is the solution of an equation representing the height of the rocket as a function of time. See Exercise 24 of Section 2.3. 80 Solving Equations Graphically Interdependence of Sections Chapter Outl... |
OLVE menu of Casio. Complete Graphs A viewing window is said to display a complete graph if it shows all the important features of the graph—including all peaks, valleys, and points where it touches an axis—and suggests the general shape of portions of the graph that are not in the window. Many different windows may sh... |
equation. If the graphs do not intersect, then they have no common output value. Therefore, there are no real solutions to the equation. The x-Intercept Method A zero of a function f is an input that produces an output of 0. For exam x3 8 23 8 0. ple, 2 is a zero of the function 2 2 Note that 2 is also a solution of t... |
zeros of f, and the f the x-intercepts 0, x 1 2 84 Chapter 2 Equations and Inequalities The x-Intercept Method Follow three steps to solve an equation by the x-intercept method. 1. Write the equation in the equivalent form f(x) 0. 2. Graph y f(x). 3. Find the x-intercepts of the graph. The x-intercepts of the graph ar... |
next page. Section 2.1 Solving Equations Graphically 85 2 This difficulty can be eliminated by using the fact that the only number whose square root is zero is zero itself. 3 3 2 Figure 2.1-3 NOTE Solving radical and rational equations is presented in Section 2.4, and radical and rational functions are presented in Ch... |
h(x) g (x). y f(x). 2. Graph 3. Find the x-intercepts of the graph of y f(x) x-intercepts of the graph of of the equation. f(x). The are the solutions The x-Intercept Method has the advantage of needing no information about the range of the functions. Applications Graphical solution methods can be helpful in dealing w... |
x 1 x3 2 5 x x2 0 0 2x2 3 2x 2 5 34. 2x3 2 2x 5 4 7. x3 4x2 10x 15 0 8. x3 9 3x2 6x 9. x4 x 3 0 In Exercises 35–40, find an exact solution of the equation in the interval shown to the right of each equation. For example, if the graphical approximation of a solu- 10. x5 5 3x4 x 11. 2x4 x3 x 3 0 tion begins.3333, check t... |
43. According to data from the U.S. Department of Health and Human Services, the cumulative number y of AIDS cases (in thousands) as of year x is approximated by y 0.062x4 1.54x3 9.21x2 57.54x 199.36 0 x 11 2 1 x 0 corresponds to 1990. During what where year did the cumulative number of cases reach 750,000? 44. a. How... |
zero, then at least one of the factors is zero. In other words, If ab 0, then a 0 or b 0 (or both). NOTE If needed, review factoring in the Algebra Appendix. Example 1 Solving a Quadratic Equation by Factoring Solve 3x2 x 10 by factoring. Solution Rearrange the terms so that one side is 0, and then factor. y 8 4 (−, 0... |
k Solve 2 x 4 1 2 2 6. Solution The equation is in the form the procedure outlined above can be applied. au2 k, where u represents x 4. Therefore ± 23 x 4 23 x 2.27 x 4 ± 23 or x 4 23 x 5.73 Divide by 2 Take square roots Subtract 4 Exact solutions Approximate solutions ■ Figure 2.2-2 y = 2(x + 4)2 y y = 6 −8 −4 −5.73 ... |
4 −8 Figure 2.2-3 2x2 6x 1 0 2x2 6x 1 x2 3x 1 2 1 2 x2 3x Subtract 1 Divide by 2 Add 2 3 2 b a 9 4 Rewrite as perfect square and simplify 7 4 A Take square root ± A 7 4 2.823 Add 3 2 or x 3 2 7 4 A 0.177 There are two real solutions. See Figure 2.2-3. ■ The technique of completing the square can be used to solve any qu... |
and with x2 8x 3 0, c 3. and apply the quadratic formula x 8 ± 282 4 2 1 1 1 2 1 8 252 2 3 2 21 8 ± 264 12 2 8 ± 252 2 0.4 or x 8 252 2 7.6. Therefore, x y 4 0 x 4 −12 −8 −4 −4 −8 −12 Figure 2.2-4 The equation has two distinct real solutions, as confirmed in Figure 2.2-4. ■ The Discriminant b2 4ac in the quadratic for... |
traits. is a polynomial is a polynomial expres- 4x3 3x2 4x 5 • no variables in denominators • no variables under radical signs As a general rule, polynomial equations of degree 3 and above are best solved by the graphical methods presented in Section 2.1. However, some equations are quadratic in form and can be solved... |
24. 2t2 11 5 In Exercises 25–28, solve the equation by completing the square. 25. x2 2x 12 26. x2 4x 30 0 27. w2 w 1 0 28. t2 3t 2 0 In Exercises 29–40, use the quadratic formula to solve the equation. 29. x2 4x 1 0 30. x2 2x 1 0 31. x2 6x 7 0 32. x2 4x 3 0 33. x2 6 2x 34. x2 11 6x 35. 4x2 4x 7 36. 4x2 4x 11 37. 4x2 8... |
. x4 2x2 1 0 63. x4 2x2 35 0 64. x4 2x2 24 0 65. 2y4 9y2 4 0 66. 6z4 7z2 2 0 67. 10x4 3x2 1 68. 6x4 7x2 3 73. Find a number k such that 4 and 1 are the solutions of x2 5x k 0. 74. Suppose a, b, and c are fixed real numbers such Let r and s be the solutions of b2 4ac 0. that ax2 bx c 0. a. Use the quadratic formula to s... |
] for b. ab 1683 ab 1683 b 1683 a [2] Divide both sides by a 98 Chapter 2 Equations and Inequalities Substitute the result into equation [1] and simplify. a b 2 41.125 [1] 1 0 1 50 Figure 2.3-1 a 1683 a 2 a 1683 41.125 Substitute 1683 a for b a 82.25 Multiply both sides by 2 6. Solve: Solve the equation by using the x-... |
the square root of both sides. h2 12.25 h ± 212.25 ± 3.5 Divide by 2 Take square root of both sides Because height is never negative, only the positive root applies to this situation. Therefore, 7. Find: Find the width by using equation [3] and h 3.5 h 3.5 in.. 8. Check: The width is twice the height and the area is c... |
h. s2 h 30,000 21.70 2 1 30,000 470.89 2 h 30,000 64.29 2 1 63.71 cm 30,000 4133.2041 2 7.26 cm Volume 63.71 21.7 2 30,000.4019 1 1 2 2 2 7.26 64.29 2 30,007.06177 2 1 1 Surface Area 2 4 21.7 6000.9180 1 2 21.7 1 21 63.71 2 2 4 64.29 6000.1857 2 1 64.29 7.26 2 2 1 1 21.7 cm 21.7 cm with a height of approxiOne base is ... |
.06s 180 s 180 0.06 3000 Therefore, the investment should be as follows: • $3000 in stock 9000 3000 • 2 1 $6000 in the savings account If this is done, the total return will be 12% of $3000 plus 6% of $6000, making a total return of $360 $360 $720 —which is 8% of $9000. ■ Distance Applications The basic formula for pro... |
around a rectangular garden that measures 24 by 40 feet. She has enough cement to cover 660 square feet. How wide should the walk be in order to use all the cement? Solution Figure 2.3-5 Let x denote the width of the walk in feet and draw a picture of the situation, as shown in Figure 2.3-5. The length of the outer re... |
y1 and Graph As shown in Figure 2.3-7, there are two points of intersection: one at approximately (2.23, 1000) and another at approximately (6.47, 1000), which is not identified on the graph. Because both are viable solutions, there are two boxes that meet the given conditions. y2 2 1000. Figure 2.3-7 11 Find the dime... |
The diameter of a circle is 16 cm. By what amount must the radius be decreased in order to decrease the area by square centimeters? 48p 8. A corner lot has dimensions 25 by 40 yards. The city plans to take a strip of uniform width along the two sides bordering the streets in order to widen these roads. How wide should... |
How much fluid should be drained and replaced with pure antifreeze so that the new mixture is 60% antifreeze? 12. A radiator contains 10 quarts of fluid, 30% of which is antifreeze. How much fluid should be drained and replaced with pure antifreeze so that the new mixture is 40% antifreeze? 13. Two cars leave a gas st... |
’s average speed. How fast does each drive? 20. To get to work Sam jogs 3 kilometers to the train, then rides the remaining 5 kilometers. If the train goes 40 km per hour faster than Sam’s constant rate of jogging and the entire trip takes 30 minutes, how fast does Sam jog? In Exercises 21–24, an object is thrown upwar... |
case? a. It is dropped from the top of a 640-foot-high building. 28. The lateral surface area of the right square S b2b2 4h2. pyramid at the right in the figure above is given by If the pyramid has height 10 feet and lateral surface area 100 square feet, what is the length of a side b of its base? Section 2.4 Other Ty... |
2 4 6 8 10 8 units Figure 2.4-1 8 Notice that property of absolute value. 5 3 0 0 0 8. 0 This is an example of a key geometric 108 Chapter 2 Equations and Inequalities Absolute Value and Distance If c and d are real numbers, then c d 00 00 is the distance between c and d on the number line. For example, the number thu... |
3 and d 5. The The Triangle Inequality For any real numbers c and d, c d 00 c 00 00 00 d 00. 00 Square Root of Squares When c is a positive number, then negative. Consider the case when 2c2 c. c 3. This is not true when c is 3 2 1 2 2 29 3, which is not 3 is equal to the absolute value of c when c is any real number. ... |
that do not involve absolute value. is either 5x 2 4x 6 x 6 4 3 2 5x 2 0 or 5x 2 x 4 1 x 4 5x 2 2 6x 2 x 2 6 1 3 Each solution must be checked in the original equation. Section 2.4 Other Types of Equations 111 x 3 2 is a solution and checks in the original equation, as shown in Figure 2.4-5. However, do not intersect ... |
3. 1 112 Chapter 2 Equations and Inequalities Power Principle CAUTION Although it is always a good idea to verify solutions, solutions to radical equations must be checked in the original equation. 5 0 0 10 Figure 2.4-7 If both sides of an equation are raised to the same positive integer power, then every solution of ... |
substitution that a solution. x 2 is an extraneous root but that x 42 is ■ Example 6 Distance Stella is standing at point A on the bank of a river that is 2.5 kilometers wide. She wants to reach point B, which is 15 kilometers downstream on the opposite bank. She plans to row downstream to point C on the opposite shor... |
with numerator f(x) and denominator g(x). As in all fractions, the denominator, g(x), cannot be zero. That is, if the frac- 0 is undefined. The following principle is used to solve fractional tion x x f 1 g 1 2 2 equations of the form 0. x x f 1 g 1 2 2 Section 2.4 Other Types of Equations 115 Solving f(x) g(x) 0 f(x)... |
distance between y and 2 is 4. 2. The distance between x and 4 is 6. 3. The distance between 3w and 2 is 8. 4. The distance between 4x and 3 is 6. 5. The distance between 2x and 4 is 5. 6. The distance between 4z and 3 is 11. 7. The distance between 3x and 2 is 5. 8. The distance between 4w and 6 is 5 2. In Exercises ... |
. 23x 2 7 31. 23 5 11x 3 32. 23 6x 10 2 33. 23 x2 1 2 34. 36. 23 x 1 2 4 1 2 2x2 5x 4 2 35. 2x2 x 1 1 37. 2x 7 x 5 23. Joan weighs 120 pounds and her doctor told her 38. 2x 5 x 1 that her weight is 5 percent from her ideal weight. What are the possible values, to the nearest pound, for Joan’s ideal body weight? 24. A t... |
a fence, 20 feet from the tree, and then to the side of a building, 35 ft from the tree, at a point 30 ft from the fence, as shown in the figure. a. If 63 ft of rope is to be used, how far from the building wall should the rope meet the fence? b. How far from the building wall should the rope meet the fence if as litt... |
because the set of real numbers is ordered. That is, for any two real numbers a and b, exactly one of the following statements is true The two statements are equivalent, and both mean that c d, d c read “c is less than or equal to d,” means either c is less than d or c is equal to d. A similar statement applies to is ... |
q, b q, b 3 1 1 1 1 Basic Principles for Solving Inequalities Solving Inequalities Solutions of inequalities in one variable are all values of the variable that make the inequality true. Such solutions may be found by using algebraic, geometric, and graphical methods, each of which is discussed in this section. Whenev... |
18 1 6 5x 6 15 1 5 7 x 7 3 Subtract 3 from each part 5 Divide each part by and reverse direction of the inequalities Intervals are usually written from the smaller to the larger, so the solution to the compound inequality is 3 6 x 6 1 5. −5 −4 −3 −2 −1 0 1 2 3 54 Figure 2.5-2 The solution of the compound inequality is... |
x x ity. The procedure for solving 2 2 and find the intervals on the x-axis where the graph is below the x-axis. A similar procedure applies when the inequality sign is reversed, except that the solution is determined by x-intervals where the graph is above the x-axis. is to graph Example 3 Solving an Inequality Solve... |
all numbers x such that 3 241 4 2.35 x 0.85 x 3 241 4 Exact solution Approximate solution ■ Example 5 Solving an Inequality Solve 21 1 0. Solution 1 f x 2 2 5, x 5 x are easily read from the facThe zeros of 1 2, and 8. Therefore, you need only determine where tored form to be the graph of f is on or below the x-axis. ... |
The graph in Figure 2.5-7 shows that the zero of f is and the graph of C is negative, i.e., below the x-axis, for values greater than 300. Therefore, to keep costs under $1600, between 300 and 450 printers should be ordered per delivery. x 300, ■ Exercises 2.5 In Exercises 1–4, express the given statement in symbols. ... |
42. x2 7x 10 0 43. x2 9x 15 0 44. x2 8x 20 0 45. 8 x x2 0 46. 4 3x x2 0 47. x3 x 0 48. x3 2x2 x 7 0 49. x3 2x2 3x 6 0 50. x4 14x3 48x2 0 51. x4 5x2 4 6 0 52. x4 10x2 9 0 53. x3 2x2 5x 7 2x 1 54. x4 6x3 2x2 6 5x 2 55. 2x4 3x3 6 2x2 4x 2 56. x5 5x4 7 4x3 3x2 2 57. 59. 61. 63. 3x 1 2x 4 7 0 x2 x 2 x2 2x 65. 2 x 3 1 x 1 5... |
hours (kwh) of electricity each month. A second freezer costs $500 and uses 100 kwh of electricity each month. The expected life of each freezer is 12 years. What is the minimum electric rate in cents per kwh for which the 12-year total cost (purchase price freezer? electricity costs) will be less for the first 126 Ch... |
week, what are the possible numbers of medallions he should make? 84. A retailer sells file cabinets for 80 x dollars each, where x is the number of cabinets she receives from the supplier each week. She pays $10 for each file cabinet and has fixed costs of $600 per week. How many file cabinets should she order from t... |
Solve 0 x4 2x2 x 2 6 11x. 0 Solution x −4 −2 0 2 4 −20 Figure 2.5.A-1 x4 2x2 x 2 The solutions of the x-intervals for which the graph of the graph of 11x. g x 0 0 6 11x f x 1 2 1 2 0 can be found be determining is below x4 2x2 x 2 0 x 0.17 A graphical intersection finder shows that the points of intersection occur and... |
3a Similarly, the inequality 5 states that r 0 0 the distance from r to 0 is greater than or equal to 5 units. These values are the numbers r such that Figure 2.5.A-3b. r 5 or r 5, as shown in −8 −6 −4 −2 0 2 4 6 8 5 units 5 units Figure 2.5.A-3b Similar conclusions hold in the general cases, with 5 replaced by any num... |
y 8 4 g(x) = x2 − x − 6 −8 −4 0 4 8 −8 −4 0 4 8 x x −4 −8 −4 −8 Figure 2.5.A-4a Figure 2.5.A-4b As shown in Figure 2.5.A-4a, the graph of the x-axis when f x 1 2 x2 x 2 is on or below 1 x 2. As shown in Figure 2.5.A-4b, the graph of above the x-axis when g x 2 1 x2 x 6 is on or x 2 or x 3. Therefore, the solutions of ... |
x2 3x 4 2 6 6 0 7 1 4x x3 0 1 4x 2 3x ` ` 6 1 x2 2 x2 2 6 1 7 4 0 0 x2 x 1 1 0 3x2 8x 2 x5 x3 1 0 6 2 0 6 2 16. 18. 20. 22. 24. 26. 0 0 0 0 x4 x3 x2 x 1 7 4 x3 6x2 4x 30. 2x2 2x 12 x3 x2 x 2 ` 7 2 32. 0 ` 0 ` x2 9 x2 4 0 x2 x 2 x2 x 2 ` 6 2 7 3 132 Chapter 2 Equations and Inequalities 33. Critical Thinking Let E be a ... |
the square............................. 92 The quadratic formula..................... 92 The discriminant.......................... 93 Polynomial equations in quadratic form....... 94 Applied problems guideline................. 97 Solutions in context....................... 98 Interest applications....................... |
other inequalities.................. 121 Quadratic and factorable inequalities......... 122 133 134 Chapter Review Section 2.5.A The intersection method................... 127 The x-intercept method.................... 127 Algebraic methods....................... 128 Important Facts and Formulas To solve an equation o... |
2 q, 1 2 10, q 2 0, q 2 0, q 2 5 Section 2.1 1. x3 2x2 11x 6; 2. x3 2x2 11x 6; 3. x4 x3 10x2 8x 16; 4. 2x4 x3 2x2 6x 2 0; 5. 6. x3 2x2 3x 4 x2 2x 15 0; 3x4 x3 6x2 2x x5 x3 2 0; 7. 2x3 2x2 3x 5 0; 8. 21 2x 3x2 4x3 x4 0; Section 2.2 9. Solve for x: 3x2 2x 5 0 10. Solve for y: 3y2 2y 5 11. Solve for z: 5z2 6z 7 12. Solve... |
? 136 Chapter Review 23. A square region is changed into a rectangular one by making it 2 feet longer and twice as wide. If the area of the rectangular region is three times larger than the area of the original square region, what was the length of a side of the square before it was changed? 24. The radius of a circle ... |
which intervals is 2 x 1 6 x? 47. Solve for x: 48. Solve for x: x 1 2 x2 1 1 2 1 x2 x 7 12. x 0. 2 49. If a. c. x 3 2x 3 x 3 2x 3 3 2x x 3 7 1, then which of these statements is true? 6 1, 7 1 6 1 2x 3 x 3 2x 3 6 x 3 b. d. e. None of these 50. If a. c. e. 0 6 r s t then which of these statements is false? b. 51. Solve... |
b. Measure the height when x has the lengths given in the chart, and cal- culate the area in each case. c. Create a scatter plot of the data. d. Estimate the length of the base that produces the maximum area, and state the approximate maximum area. Solution a. The base must be greater than 0 and less than 8.5 inches, ... |
of x and the area, differential calculus is needed. However, graphing technology can provide very good approximations. Exercises In each problem, find the maximum by using a numerical method like the one shown in Example 1, and then by using an analytical and graphical method like the one shown in Example 2. Answer al... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.