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stem. For example, point (4, 3), which lies in the region, satisfies the system because its x-value satisfies one of the given inequalities, 4 2, and its y-value satisfies the other inequality, 3 2. EXAMPLE 2 Graph the solution set of 3 x 5 in a coordinate plane. Solution The inequality 3 x 5 means 3 x and x 5. This ma... |
2x x 4 36. 2x y 4 x y 2 In 37β41, for each problem, write two equations in two variables and solve algebraically. 37. The sum of two numbers is 7. Their difference is 18. Find the numbers. 38. At a store, 3 notebooks and 2 pencils cost $2.80. At the same prices, 2 notebooks and 5 pencils cost $2.60. Find the cost of a... |
280 2 2 7 2 5 or 280 23 5 7 Let us find the greatest common factor of 180 and 54. 180 2 2 3 3 5 β β β 54 2 3 3 3 β β 3 3 β Greatest common factor 2 or or or 22 32 5 2 33 2 32 or 18 Only the prime numbers 2 and 3 are factors of both 180 and 54. We see that the greatest number of times that 2 appears as a factor of both... |
. (3a 4b)(3a 4b) Think (y)2 (7)2 (3a)2 (4b)2 Write y2 49 9a2 16b2 EXERCISES Writing About Mathematics 1. Rose said that the product of two binomials is a binomial only when the two binomials are the sum and the difference of the same two terms. Miranda said that that cannot be true because (5a 10)(a 2) 5a2 20, a binomi... |
) (1)(12) (2)(6) (2)(6) (3)(4) (3)(4) (3) The possible factors are: (y l)(y 12) (y l)(y 12) (y 2)(y 6) (y 2)(y 6) (y 3)(y 4) (y 3)(y 4) (4) When we find the middle term in each of the trinomial products, we see that only for the factors (y 6)(y 2) is the middle term 8y: 6y |____| (y 6)(y 2) β 2y 6y 8y |____________| 2y... |
l and a factor that is a perfect square? (1) a2y 10ay 25y (2) 2ax2 2ax 12a (3) 18m2 24m 8 (4) c2z2 18cz2 81z2 34. Of the four polynomials given below, explain how each is different from the others. x2 9 x3 5x2 6x 35. If the length and width of a rectangle are represented by 2x 3 and 3x 2, respectively, express the area... |
number is one of the three equal factors of the number. For example, 2 is a cube root of 8 because 2 2 2 8, or 23 8. A cube root of 8 is written as 3 8 . " Finding a cube root of a number is the inverse operation of cubing a ) if and only number. In general, the cube root of b is x (written as if x3 b. 3 b 5 x " 3 28 ... |
" " Is 56 " a rational or an irrational number? Solution Since 56 is a positive integer that is not a perfect square, there is no rational number that, when squared, equals 56. Therefore, is irrational. 56 " Calculator Solution STEP 1. Evaluate 56 . ENTER: Β― 56 ENTER " 2nd DISPLAY STEP 2. To show that the calculator d... |
he approximations in parts a and c equal? 27. 300 " 28. 180 " 29. 2 288 " 30. 1 3 252 " Addition and Subtraction of Radicals 487 ? Explain your answer. 31. a. Does 9 1 16 5 9 1 16 b. Is finding a square root distributive over addition? " " 32. a. Does 169 2 25 5 169 2 25 ? Explain your answer. b. Is finding a square ro... |
whether each quotient is a rational number or an irrational number. 19. 5 " 7 23. 9 16 " " 20. 24. 50 2 18 25 " " " " 21. 25. 18 3 " " 25 " 5 " 24 2 22. 49 " 7 " 26. 3 " 6 " 54 3 496 Operations With Radicals In 27β34, simplify each expression. Assume that all variables represent positive numbers. 27. 31. 36 49 $ 10 8 ... |
s 8. a point on the x-axis. The slope of a. Write the equation of g AB . b. Draw g AB on graph paper. c. What are the coordinates of B? d. What is the x-intercept of g ?AB CHAPTER 13 CHAPTER TABLE OF CONTENTS 13-1 Solving Quadratic Equations 13-2 The Graph of a Quadratic Function 13-3 Finding Roots from a Graph 13-4 Gr... |
n set, each different value of x is paired with one and only one value of y. The graph of any quadratic function is a parabola. Because the graph of a quadratic function is nonlinear, a larger number of points are needed to draw the graph than are needed to draw the graph of a linear function. The graphs of two equatio... |
Make a table of y x 0 1 2 3 4 5 6 x2 6x 0 0 1 6 4 12 9 18 16 24 25 30 36 36 1 β1 1 x c. The maximum value of the area, y, is 9. Note: The graph shows all possible values of x and y. Since both the measure of a side of the rectangle, x, and the area of the rectangle, y, must be positive, 0 x 6 and 0 y 9. Since (2, 8) i... |
ratic Relations and Functions EXAMPLE 2 Use the graph of y x2 3x 4 to find the linear factors of x2 1 3x 2 4 . y Solution The graph intersects the x-axis at (4, 0) and (1, 0). Therefore, the roots are 4 and 1. If x 4, then x (4) (x 4) is a factor. If x 1, then (x 1) is a factor. O 1 β1 β1 x Answer The linear factors of... |
5 a 2b 2 b 5 0 b 5 b 5 a 2b 2 a 2 233 5 A 256 5 1 2 B a 2(5) 2 a 12 532 Quadratic Relations and Functions EXERCISES Writing About Mathematics 1. Explain why the equations y x2 and y 4 have no common solution in the set of real numbers. 2. Explain why the equations x2 y2 49 and x 8 have no common solution in the set of... |
6. Jean Forester has a small business making pies and cakes. Today, she must make at least 4 cakes to fill her orders and at least 3 pies. She has time to make a total of no more than 10 pies and cakes. a. Let x represent the number of cakes that Jean makes and y represent the number of pies. Write three inequalities t... |
5x x 2 5 ? Explain your answer. x2 2 5x x 2 5 x x to become x(x2 2 5x) x(x 2 5) , will it be equivalent to 14-3 MULTIPLYING FRACTIONS The product of two fractions is a fraction with the following properties: 1. The numerator is the product of the numerators of the given fractions. 2. The denominator is the product of ... |
that the LCD is the smallest possible common denominator. Procedure To add (or subtract) fractions that have different denominators: 1. Choose a common denominator for the fractions. 2. Change each fraction to an equivalent fraction with the chosen common denominator. 3. Write a fraction whose numerator is the sum (or ... |
one-fifth of a positive number and one-tenth of that number is 10. Find the number. 40. If one-half of a number is increased by 20, the result is 35. Find the number. 41. If two-thirds of a number is decreased by 30, the result is 10. Find the number. 42. If the sum of two consecutive integers is divided by 3, the quot... |
st, at 7% interest. If the total annual interest from these two investments is at least $160, what is the smallest amount he could have invested at 71 2% ? 37. Mr. Lehtimaki wanted to sell his house. He advertised an asking price, but knew that he would accept, as a minimum, nine-tenths of the asking price. Mrs. Patel ... |
ne-half of the result obtained when 5 is subtracted from the number. Find the number. 37. Of the total number of points scored by the winning team in a basketball game, one-fifth was scored in the first quarter, one-sixth was scored in the second quarter, one-third was scored in the third quarter, and 27 was scored in ... |
dy. In an empirical study, we perform an experiment many times, keep records of the results, and then analyze these results. 20 20 20 20 20 20 20 20 20 20 Number of Heads Number of Tosses For example, ten students decided to take turns tossing a coin. Each student completed 20 tosses and the number of heads was recorde... |
are equal in size and are numbered 1, 2, 3, and 4. An experiment was conducted by five people to find the probability that the arrow will land on the 2. Each person spun the arrow 100 times. When the arrow landed on a line, the result did not count and the arrow was spun again. a. Before doing the experiment, what pro... |
2 An aquarium at a pet store contains 8 goldfish, 6 angelfish, and 7 zebrafish. David randomly chooses a fish to take home for his fishbowl. a. How many possible outcomes are in the sample space? b. What is the probability that David takes home a zebrafish? Solution a. Each fish represents a distinct outcome. Therefore... |
t least 10 cents? c. exactly 3 cents? d. more than 3 cents? Solution The sample space for this example is {N, D1, D2, Q}. Therefore, n(S) 4. a. There are two coins worth exactly 10 cents, D1 and D2. Therefore, n(E) 2 and P(coin worth 10 cents) n(E) n(S) 5 2 4 5 1 2 . b. There are three coins worth at least 10 cents, D1... |
lling a number less than 2. Then: P(C) n(C) n(S) 5 1 6 Now, what is the probability of obtaining a number on the die that is even or less than 2? We may think of this as event (A or C). For the example above, there are four outcomes in the event (A or C): 1, 2, 4, and 6. Each of these numbers is either even or less tha... |
A) In rolling a fair die, we know that P(4) since there is only one outcome for the event (rolling a 4). We can also say that P(not 4) since there are five outcomes for the event (not rolling a 4): 1, 2, 3, 5, and 6. 5 6 5 1 6 We can think of these probabilities in another way. The event (4) and the event (not 4) are m... |
in either of two ways. By the counting principle, the sample space contains 2 2 2 or 8 possible outcomes. By letting H represent a head and T represent a tail, we can illustrate the sample space by a tree diagram or by a list of ordered triples: We did not attempt to draw a graph of this sample space because we would n... |
lowed by 3 letters c. 4 numbers, followed by 2 letters 22. In a school cafeteria, the menu rotates so that P(hamburger) , P(apple pie) , and 4 5 P(soup) . The selection of menu items is random so that the appearance of hamburgers, apple pie, and soup are independent events. On any given day, what is the probability tha... |
arbles of which two are blue. On the first draw: P(blue) 2 6 Since the blue marble drawn is not replaced, five marbles, of which only one is blue, are left in the jar. On the second draw: P(blue given that the first was blue) 1 5 Then: P(both blue) 2 6 3 1 5 5 2 30 or 1 15 Answer c. If both marbles are the same color, ... |
d was a quarter. 16. One hundred boys and one hundred girls were asked to name the current Secretary of State. Thirty boys and sixty girls knew the correct name. One of these boys and girls is selected at random. a. What is the probability that the person selected knew the correct name? b. What is the probability that ... |
n also be written as: nPr n(n 1)(n 2) (n r 1) Note that when there are r factors, the last factor is (n r 1). In the example given above, in which three students were selected from a group of six, n 6, r 3, and the last factor is n r 1 6 3 1 4. Permutations and the Calculator There are many ways to use a calculator to ... |
f Marieβs days. Number of arrangements Answer 210 possible schedules 7! 3! 3 2! 3 2 210 638 Probability EXERCISES Writing About Mathematics 1. a. List the six different arrangements or permutations of the letters in the word TAR. b. Explain why exactly six arrangements are possible. 2. a. List the three different arran... |
y 1 line exists, namely, . Since order is not important here, this is a combination of 6 points, taken g AB 2 at a time. Answer 15 lines 6C2 6 P 2 15 644 Probability EXAMPLE 5 Lisa Dwyer is a teacher at a local high school. In her class, there are 10 boys and 20 girls. Find the number of ways in which Ms. Dwyer can sel... |
els (2) 1 caramel and 3 creams (3) 2 caramels and 2 creams (4) no caramels (5) a second cream given that 2 caramels and a cream have been selected 7. Two cards are drawn at random from a 52-card deck without replacement. Find the proba- bility of drawing: a. the ace of clubs and jack of clubs in b. a red ace and a blac... |
, 6, 7, and 8 if repetition is not allowed? b. What is the probability that such a 3-digit numeral is greater than 400? 19. a. If a 4-member committee is formed from 3 girls and 6 boys in a club, how many committees can be formed? b. If the members of the committee are chosen at random, find the proba- bility that the ... |
ata to keep the public informed on the progress of the candidates. Election campaigns are just one example of the use of statistics to organize data in a way that enables us to use available information to evaluate the current situation and to plan for the future. CHAPTER 16 CHAPTER TABLE OF CONTENTS 16-1 Collecting Da... |
ERIOD 200.0 199.9 199.8 199.7 199.6 199.5 199.4 199.3 199.2 199.1 199. Collecting Data 665 POPULATION OF ANYTOWN, U.S ( 500 400 300 200 100 Month Year EXERCISES Writing About Mathematics 1. A census attempts to count every person. Explain why a census may be unreliable. 2. A sample of a new soap powder was left at each... |
s, 0 and 5, the stem is 0; for the other data values, the stem is 1, 2, 3, or 4. Then the units digit will be the leaf. We construct the diagram as follows: STEP 1. List the stems, starting with 4, under one another Stem Leaf to the left of a vertical line beneath a crossbar. STEP 2. Enter each score by writing its lea... |
bel intervals of equal length on a horizontal scale. Label the horizontal scale, telling what the numbers represent. Interval Frequency 16β19 20β23 24β27 28β31 32β35 36β39 40β43 5 11 8 5 7 3 1 (3) Draw the bars vertically, leaving no gaps between the intervals 12 11 10 16β19 20β23 24β27 28β31 32β35 36β39 40β43 Mileage ... |
n by the mean or the median. 3. The number of photographs printed from each of Reneeβs last six rolls of film are 8, 8, 9, 11, 11, and 12. Since 8 appears twice and 11 appears twice, we say that there are two modes: 8 and 11. We do not take the average of these two numbers since the mode tells us where most of the scor... |
e obtain on the next test so that her average for the five tests will be 90? 27. The first three test scores are shown below for each of four students. A fourth test will be given and averages taken for all four tests. Each student hopes to maintain an average of 85. Find the score needed by each student on the fourth ... |
uped using intervals of length other than 1, there is no simple procedure to identify the interval containing the mean. However, the mean can be approximated by assuming that the data are equally distributed throughout each interval. The mean is then found by using the midpoint of each interval as the value of each ent... |
h and 8th values. Therefore, 8.5 is the second quartile. Median 8 1 9 2 8.5 (3) Find the first quartile. There are seven values less than 8.5. The middle value is the 4th value from the lower end of the set of data, 6. Therefore, 6 is the first, or lower, quartile. (4) Find the third quartile. There are seven values gr... |
rank: 0.63 192 15 for the data value $2.75. β 63% Answers a. Diagram b. median $2.57 c. first quartile $2.48; third quartile $2.80 d. Diagram e. Diagram f. 63rd percentile Note: A cumulative frequency histogram can be drawn on a calculator just like a regular histogram. In list L2, where we previously entered the frequ... |
hrough different areas on his trip, his average speed and the length of time he drives each day vary. The chart below shows a record of average speed and time for a 10-day period. Speed Time 50 10 64 68 60 54 66 70 62 64 58 7.9 7.5 8.5 9.0 7.0 7.1 8.0 8.2 9.0 11 10.5 10 9.5 9 8.5 8 7.5 7 6. 50 55 60 65 70 75 Speed in M... |
of gas purchased and the number of miles driven since the last fill-up. Her record for the first 2 months is as follows: Gallons of gas 10 12 9 6 11 10 8 12 10 7 Miles driven 324 375 290 190 345 336 250 375 330 225 a. Draw a scatter plot of the data. Let the horizontal axis represent the number of gallons of gas and t... |
87, 75, 77 a. Use a stem-and-leaf diagram to organize the data. b. Draw a histogram, using 50β59 as the lowest interval. c. Draw a cumulative frequency histogram. d. Draw a box-and-whisker plot. 6. The weights, in kilograms, of five adults are 53, 72, 68, 70, and 72. a. Find: (1) the mean (2) the median (3) the mode b... |
59 linear pair of, 252 measuring, 248 obtuse, 249 opposite, 272 pairs of, 250β254 perpendicular, 249 parallel lines and, 258β261 parallelograms and, informal proofs, 274β275 polygons and, 275 of quadrilaterals, 273 right, 248 sides of an, 248 sine of, 317 straight, 249 sum of the measures of, for polygons, 275 for tria... |
face area, 293 transforming, 143β145 for volume, 293 writing, 107β108 Fractional coefficients solving equations with, 556β559 solving inequalities with, 562β563 Fractional equation(s), 565 solving, 565β567 Fractional expression, 540 Fraction line, as grouping symbol, 41 Fraction(s) addition of, 550β554 algebraic, 540 c... |
es with replacement, 617β618 without replacement, 617 uniform, 586 writing as fractions, 586 Problem solving formulas in, 134β137 inequalities in, 157β159 tangent ratio in, 313β315 trigonometric ratios in, 327β328 Product, 443 Properties of equality, 117β119 addition property of equality, 118 division property of equal... |
gle of depression, 313 angle of elevation, 313 in problem solving, 313β315 finding, on a calculator, 308β311 Terminating decimals, 14, 586 Term(s), 95, 122, 168 factors of, 95 like, 123, 168 lowest, 212 multiplication of the sum and difference of two, 450β451 reducing fractions to lowest, 541β543 similar, 123, 168 unde... |
iew Note: Persian mathematician Omar Khayyam would solve algebraic problems geometrically by intersecting graphs rather than solving them algebraically. 35 1.2 Practice - Two-Step Problems Solve each equation. 1) 5 + n 4 = 4 3) 102 = 7r + 4 β 8n + 3 = 5) β 7) 0 = 6v β 8 = x 9) β 11 β 13) 12 + 3x = 0 β 15) 24 = 2n 8 β 1... |
= 5 6 Same problem, with common denominator 6 + 21 6 + 21 6 Add 21 6 to both sides 3 4 3 4 x = x = 26 6 13 3 Reduce 26 6 to 13 3 Focus on multiplication by 3 4 We can get rid of 3 4 by dividing both sides by 3 same as multiplying by the reciprocal, so we will multiply both sides by 4 3. 4. Dividing by a fraction is th... |
inator Focus on the positive 2 Subtract 2 from both sides 2 Still need to clear the negative (2 β b)a = (2 β A 2 b β b)A (2 β b 2 β b))( ( β β b) = ( 1) β 1) Multiply (or divide) each term by 1 β Our Solution Both answers to the last two examples are correct, they are just written in a different form because we solved ... |
direct variation or directly proportional. If we see this phrase in the problem we know to divide to ο¬nd the constant of variation. Example 92. m is varies directly as n β²β²Directlyβ²β² tells us to divide = k Our formula for the relationship m n In kickboxing, one will ο¬nd that the longer the board, the easier it is to b... |
stop in 72 ft? 34. The drag force on a boat varies jointly as the wetted surface area and the square of the velocity of a boat. If a boat going 6.5 mph experiences a drag force of 86 N when the wetted surface area is 41.2 ft2, how fast must a boat with 28.5 ft2 of wetted surface area go in order to experience a drag f... |
o they each cost? Love Seat x With no information about the love seat, this is our x Sofa 2x Sofa is double the love seat, so we multiply by 2 S + L = 444 Together they cost 444, so we add. (x) + (2x) = 444 Replace S and L with labeled values 3x = 444 Parenthesis are not needed, combine like terms x + 2x 3 x = 148 Our ... |
Divide both sides by 2 Our solution for x β 5 Age Now Caremen 13 + 12 = 25 David 13 Replace x with 13 to answer the question Carmen is 25 and David is 13 Sometimes we are given the sum of their ages right now. These problems can be tricky. In this case we will write the sum above the now column and make the ο¬rst perso... |
oth times The distance column is ο¬lled in by multiplying rate by time We have total distance, 8 miles, under distance The distance column gives equation by adding Combine like terms, 4t + 6t Divide both sides by 10 Our solution for t, 4 5 hour (48 minutes) As the example illustrates, once the table is ο¬lled in, the equ... |
The sprinter took 55 s to run to the end of the track and jog back. Find the length of the track. 21. A motorboat leaves a harbor and travels at an average speed of 18 mph to an island. The average speed on the return trip was 12 mph. How far was the island from the harbor if the total trip took 5 h? 22. A motorboat l... |
each side by 2 β β 3y = 6 Next x = 3 3y = 6 Multiply β β 6 β 3y = 0 3 3 β y = 0 β β Subtract 9 from both sides Divide each side by 3 β Solution for y when x = 3, add this to table β 92 Our completed table. 3, ( β β 4), (0, 2), (3, 0) Table becomes points to graph Graph points and connect dots Our Solution 93 2.1 Pract... |
ple 136. x 1 2 y = mx + b β Graph y = 4 Recall the slope intercept formula β Idenο¬ty the slope, m, and the y intercept Make the graph Starting with a point at the y-intercept of 4, β Then use the slope rise run, so we will rise 1 unit and run 2 units to ο¬nd the next point. Once we have both points, connect the dots to ... |
cian Euclid lived around 300 BC and published a book titled, The Elements. In it is the famous parallel postulate which mathematicians have tried for years to drop from the list of postulates. The attempts have failed, yet all the work done has developed new types of geometries! As the above graphs illustrate, parallel... |
one side Subtract 10x from both sides 20 Add 20 to both sides β + 20 8 < 2x Divide both sides by 2 2 4 < x Be careful with graph, x is larger! 2 (4, ) β Interval Notation It is important to be careful when the inequality is written backwards as in the previous example (4 < x rather than x > 4). Often students draw the... |
th sides by 3 Dividing by a negative switches the symbol β β > 4 Absolute value is greater, use OR β 3 | 129 x > 4 OR x 6 4 Graph β ( 4] , β βͺ [4, ) β β β Interval Notation In the previous example, we cannot combine terms, the β adding 4, then divide by 3 has an absolute value attached. So we must ο¬rst clear the 3. The... |
3 Subtract y = 7 We now also have y (5, 7) Our Solution When we know what one variable equals we can plug that value (or expression) in for the variable in the other equation. It is very important that when we substitute, the substituted value goes in parenthesis. The reason for this is shown in the next example. Examp... |
r solution for x 3(5) + 6y = 15 + 6y = 15 9 Plug into either original equation, simplify β 9 β 15 β Subtract 15 from both sides β 147 6y = 6 β y = (5, β β 24 Divide both sides by 6 6 4 Now we have our solution for y 4) Our Solution It is important for each problem as we get started that all variables and constants are ... |
= β 4) x + y + z = 2 1 4y + 5z = 31 6x 5x + 2y + 2z = 13 β 3) 3x + y β 5) x + 6y + 3z = 4 2x + y + 2z = 3 2y + z = 0 3x β 7 β β β 2y + 3z = 6 β 2x x 3 z = 0 9 + 2z = 0 β 11) 2x + y 3z = 1 β β x 4y + z = 6 4x + 16y + 4z = 24 β 3z = 0 13) 2x + y β x 4y + z = 0 4x + 16y + 4z = 0 β 15) 3x + 2y + 2z = 3 x + 2y 2x z = 5 4y +... |
Subtract 320 from both sides Divide both sides by 13000 x = 0.07 We have our x, 7% interest (0.07) + 0.04 0.11 S5000 at 7% and S8000 at 11% Second account is 4% higher The account with S8000 is at 11% Our Solution 163 4.5 Practice - Value Problems Solve. 1) A collection of dimes and quaters is worth S15.25. There are 1... |
another coο¬ee that costs S1.50 should the cafe mix with the ο¬rst? Amount Part Total Start Add Final 40 x 3 1.5 Amount Part Total Start Add Final 40 x 40 + x 3 1.5 2.5 Set up mixture table. We know the starting amount and its cost, S3. The added amount we do not know but we do know its cost is S1.50. Add the amounts to ... |
is 85% silk. How many kilograms of each must be woven together to make 75 kg of cloth that is 96% silk? 33) A carpet manufacturer blends two ο¬bers, one 20% wool and the second 50% wool. How many pounds of each ο¬ber should be woven together to produce 600 lb of a fabric that is 28% wool? 34) How many pounds of coο¬ee tha... |
using the properties of exponents. There are a few special exponent properties that deal with exponents that are not positive. The ο¬rst is considered in the following example, which is worded out 2 diο¬erent ways: Example 214. a3 a3 Use the quotient rule to subtract exponents a0 Our Solution, but now we consider the pro... |
.1) = 28.67 Multiply numbers Γ 10110β Γ 101 Convert this number into scientiο¬c notation 2.867 3109 = 107 Use product rule, add exponents, using 101 from conversion 2.867 107 Our Solution Γ World View Note: Archimedes (287 BC - 212 BC), the Greek mathematician, developed a system for representing large numbers using a s... |
binomial. The letters of FOIL help us remember every combination. F stands for First, we multiply the ο¬rst term of each binomial. O stand for Outside, we multiply the outside two terms. I stands for Inside, we multiply the inside two terms. L stands for Last, we multiply the last term of each binomial. This is shown in... |
5) β 39) (4k + 2)2 2) (a 4) (x β β 4)(a + 4) 3)(x + 3) 6) (8m + 5)(8m 5) β 3) 8) (2r + 3)(2r β 7)(b + 7) 10) (b β 12) (7a + 7b)(7a 7b) β 3x)(3y + 3x) 14) (3y β 16) (1 + 5n)2 18) (v + 4)2 20) (1 β 22) (7k 6n)2 β 7)2 5)2 24) (4x β 26) (3a + 3b)2 28) (4m n)2 β 30) (8x + 5y)2 32) (m 7)2 β 34) (8n + 7)(8n 36) (b + 4)(b β 38... |
e are variables in our problem we can ο¬rst ο¬nd the GCF of the num- 212 bers using mental math, then we take any variables that are in common with each term, using the lowest exponent. This is shown in the next example Example 263. GCF of 24x4y2z, 18x2y4, and 12x3yz5 24 6 = 4, 18 6 = 3, 12 6 = 2 Each number can be divid... |
d View Note: Soο¬a Kovalevskaya of Russia was the ο¬rst woman on the editorial staο¬ of a mathematical journal in the late 19th century. She also did research on how the rings of Saturn rotated. 219 6.2 Practice - Grouping Factor each completely. 1) 40r3 8r2 β β 25r + 5 9n + 6 3) 3n3 2n2 β β 5) 15b3 + 21b2 35b 49 β 7) 3x3... |
ethod becomes easier. Example 298. 18x3 + 33x2 3x[6x2 + 11x 4x β β β β 2(2x + 5)] 30x GCF = 3x, factor this out ο¬rst 10] Multiply to ac or (6)( 10) = β 10] The numbers are 15 and Factor by grouping β 3x[6x2 + 15x 3x[3x(2x + 5) β 3x(2x + 5)(3x 2) Our Solution β 60, add to 11 4, split the middle term β As was the case wi... |
316. 5 + 625y3 GCF ο¬rst, 5 5(1 + 125y3) Two terms, sum of cubes 5(1 + 5y)(1 β 5y + 25y2) Our Solution It is important to be comfortable and conο¬dent not just with using all the factoring methods, but decided on which method to use. This is why practice is very important! 235 6.6 Practice - Factoring Strategy Factor ea... |
to aid in the use of their base 20 system as a place holder! Rational expressions are easily evaluated by simply substituting the value for the variable and using order of operations. Example 327. x2 4 β x2 + 6x + 8 when x = 6 β Substitute β 5 in for each variable 6)2 ( β 6)2 + 6( ( β 36 36 + 6( 4 6) + 8 4 6) + 8 β β ... |
ready in the denominator. Example 342. 5a 3a2b = ? 6a5b3 Idenο¬ty what factors we need to match denominators 2a3b2 3 Β· 2 = 6 and we need three more aβ²s and two more bβ²s 5a 3a2b 2a3b2 2a3b2 Multiply numerator and denominator by this 10a4b2 6a5b3 Our Solution Example 343. x 2 β x + 4 = ? x2 + 7x + 12 (x + 4)(x + 3) Factor... |
x + 1 x Our Solution The process is the same if the LCD is a binomial, we will need to distribute Multiply each term by LCD, (x + 4) 263 3(x + 4) 2(x + 4) x + 4 β 5(x + 4) + 2(x + 4) x + 4 Reduce fractions 2(x + 4) 3 5(x + 4) + 2 β Distribute 2x 3 8 5x + 20 + 2 β β Combine like terms 2x 5 β β 5x + 22 Our Solution The m... |
= Our Solution 8 19 8 We will use the same process to solve rational equations, the only diο¬erence is our 274 LCD will be more involved. We will also have to be aware of domain issues. If our LCD equals zero, the solution is undeο¬ned. We will always check our solutions in the LCD as we may have to remove a solution fr... |
76. A child is perscribed a dosage of 12 mg of a certain drug and is allowed to reο¬ll his prescription twice. If a there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 reο¬lls)? Convert 3 Rx to doses 1 Rx = 60 tab, 1 tab = 4 mg, 1 dose = 12mg Identify ... |
8.1 Practice - Square Roots Simplify. 1) 245β 3) 36β 5) 12β 7) 3 12β 9) 6 128β 11) 13) 8 392β β 192nβ 15) β 196v2 17) β 252x2 19) 21) 23) β 100k4 7 64x4β 5 36mβ β β β 25) 45x2y2 p p 27) 29) 16x3y3 320x4y4 p 31) 6 80xy2 p 33) 5 245x2y3 35) 37) p β β 2 180u3v 8 180x4y2z4 β 39) 2 p 80hj4k 41) p 4 β p 54mnp2 2) 125β 4) 19... |
or like terms have been combined. Division with radicals is very similar to multiplication, if we think about division as reducing fractions, we can reduce the coeο¬cients outside the radicals and reduce the values inside the radicals to get our ο¬nal solution. Quotient Rule of Radicals: a c bmβ dmβ = a c b d m r Example... |
implify radicals with exponents, we divide the exponent by the index. Another way to write division is with a fraction bar. This idea is how we will deο¬ne rational exponents. Deο¬nition of Rational Exponents: a n m = ( amβ )n The denominator of a rational exponent becomes the index on our radical, likewise the index on ... |
. The Mayans of Central America later made up the number zero when they found use for it as a placeholder. Ancient Chinese Mathematicians made up negative numbers when they found use for them. In mathematics, when the current number system does not provide the tools to solve the problems the culture is working with, we... |
( 7x + 2 )2 = 42 7x + 2 = 16 2 2 = 4 Even index! We will have to check answers Square both sides, simplify exponents Solve Subtract 2 from both sides β β 7x = 14 Divide both sides by 7 7 7 x = 2 Need to check answer in original problem 7(2) + 2 β 14 + 2 = 4 Multiply = 4 Add p 16β = 4 Square root 4 = 4 True! It works! ... |
ed the even root property, or to see if we need to check our answer because there was an even root in the problem. When checking we will usually want to check in the radical form as it will be easier to evaluate. Β± 335 9.2 Practice - Solving with Exponents Solve. 1) x2 = 75 3) x2 + 5 = 13 5) 3x2 + 1 = 73 7) (x + 2)5 = ... |
ly one solution if the square root simpliο¬es to zero. Example 469. 12 x = Β± 4x2 ( β 12)2 β p 2(4) β 12 x = Β± 12x + 9 = 0 4(4)(9) a = 4, b = β 12, c = 9, use quadratic formula Evaluate exponents and multiplication 144 β Evaluate subtraction inside root 0β Evaluate root β 144 8 12 x = x = Β± 8 12 0 Β± 8 12 x = 8 3 2 x = Ev... |
6β 2 x = Β± 3 2 Β± r = x2β or i 2β , Β± Our Solution 353 or y = 2 We have y, still need x. Substitute into y = x2 Square root of each side Simplify each root, rationalize denominator When we create a new variable for our substitution, it wonβt always be equal to just another variable. We can make our substitution variabl... |
a of center, length times width 80000 Β· 2x)(200 β (400 β 800x β 4x2 β β 80000 400x + 4x2 = 57600 FOIL 1200x + 80000 = 57600 Combine like terms 57600 57600 Make equation equal zero β β 4x2 β 4(x2 4(x 1200x + 22400 = 0 300x + 5600) = 0 280)(x 20) = 0 β 280 = 0 or x 20 = 0 β + 20 + 20 β β β + 280 + 280 x Factor out GCF of... |
ntice: x, Total Reduce and convert to fraction Clearly deο¬ne variables Using reciprocals, make equation 1(6x(x x β β 1 1)) + 1(6x(x x β 1)) = 5(6x(x 6 β 1) Multiply each term by LCD 6x(x 1) β 366 6x + 6(x 1) = 5x(x 6 = 5x2 6 = 5x2 β β β β 6x + 6x 12x 12x + 6 0 = 5x2 β β β 1) Reduce each fraction 5x Distribute 5x Combin... |
14! 371 9.9 Practice - Simultaneous Product Solve. 1) 3) xy = 72 (x + 2)(y β xy = 150 4) = 128 6)(y + 1) = 64 (x β 5) xy = 45 (x + 2)(y + 1) = 70 7) xy = 90 5)(y + 1) = 120 (x β 9) 11) xy = 12 (x + 1)(y β xy = 45 4) = 16 5)(y + 3) = 160 (x β 2) 4) 6) 8) 10) (x xy = 180 1)(y β β xy = 120 1 2) = 205 (x + 2)(y 3)=120 β x... |
hen driving 120 kilometers at a rate that is 20 km/hr faster. At what rates should he drive if he plans to complete the test in 3 1 2 hours? 10) A train traveled 240 kilometers at a certain speed. When the engine was replaced by an improved model, the speed was increased by 20 km/hr and the travel time for the trip was... |
his deο¬nition is to look at the graphs of a few relationships. Because x values are vertical lines we will draw a vertical line through the graph. If the vertical line crosses the graph more than once, that means we have too many possible y values. If the graph crosses the graph only once, then we say the relationship ... |
swer into the outer function. This is shown in the following example. 395 Example 521. a(x) = x2 2x + 1 β b(x) = x β b)(3) Find (a β¦ 5 Rewrite as a function in function a(b(3)) Evaluate the inner function ο¬rst, b(3) 2 This solution is put into a, a( 5 = 2) β b(3) = (3) 2)2 β 2( β a( β 2) = ( 2) + 1 Evaluate β a( β β 2)... |
(x) = x + 1 x + 2 3 14) f (x) = β x 3 β 16) g(x) = 9 + x 3 18) f (x) = 5x 20) f (x) = 12 β 2 β 4 15 3x 22) g(x) = β 5 x + 2 2 2x q 3 24) f (x) = β β x + 3 26) h(x) = x x + 2 28) g(x) = β 30) f (x) = 5x x + 2 3 5 β 4 32) f (x) = 3 34) g(x) = (x β 2x5 1)3 + 2 36) f (x) = β β 1 x + 1 37) f (x) = 7 β x β x 39) g(x) = 3x 2 ... |
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