Reteaching Comparing Direct and Inverse Variation

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1 Name Date Class Comparing Direct and Inverse Variation INV 7 You have explored direct variation and inverse variation. Now you will compare these two types of variation. An equation that relates the distance d an object moves, the time t the object takes to move that distance, and the rate r at which the object moves is d r t. This is a direct variation equation if r is a constant. In 1973, the racehorse Secretariat set a world record for a. kilometer race while racing in the Belmont Stakes, the third race of the Triple Crown series. He ran the race at an average speed of about kilometers per hour. Assume that Secretariat can run at half this speed, 3 kilometers per hour, for any length of time. Identify the constant of variation. The direct variation equation is d r t. r represents the rate (speed) and is the constant of variation. k 3 Write an equation of direct variation that Use the equation d r t. r is 3. relates Secretariat s time to distance traveled. d 3t Make a table of values for the distance Time (hr) 1 3 Secretariat can run in 1,, and 3 hours. Distance (km) 3 9 Make a graph of the values in the table above. Distance (km) Time (hr) Extend the table and graph shown above. 1. Find the distance traveled in hours and. Graph the values from Problem 1. 5 hours. Add the values to the table. Use d r t. d 3 1 d Time (hr) Distance (km) Saxon. All rights reserved. 153 Saxon Algebra 1 Distance (km) Time (hr)

2 continued INV 7 If d is a constant and r and t are variables, then d r t becomes an inverse variation equation. It can be rewritten as r d t. Now assume that Secretariat trains by running kilometers every week. His average weekly speed will vary with the time he spends running. Identify the constant of variation. Write an equation of inverse variation that relates Secretariat s speed to his time spent running. Make a table of values for Secretariat s speed if he runs km in 1,, and 3 hours. The inverse variation equation is d r t. d represents the distance and is the constant of variation. k Use the equation d r t. d is. rt, or r t. Time (hr) 1 3 Speed (km/hr) 3 Make a graph of the values in the table above. Speed (km/hr) Time (hr) Extend the table and graph shown above. 3. Find Secretariat s speed if he runs km. Graph the values from Problem 3. in hours and 5 hours. Add the values to the table. Use r t. r 15 r 5 1 Time (hr) Speed (km/hr) Speed (km/hr) Time (hr) Identify the constant of variation. Then write the equation of variation. 5. y varies directly with x; y 18 when x 3 k ; y x. s varies inversely with t; s 1 when t 3 k 8; s 8 t Saxon. All rights reserved. 15 Saxon Algebra 1

3 Name Date Class Making and Analyzing Scatter Plots 71 A scatter plot relates two sets of data with plotted ordered pairs. Correlation is one way to describe the relationship between two sets of data. Positive Correlation y Data: As one set increases, the other set increases. Graph: The graph goes up from left to right. Negative Correlation y Positive x Data: As one set increases, the other set decreases. Graph: The graph goes down from left to right. Negative x No Correlation Data: There is no relationship between the sets. y Graph: The graph has no pattern. No x Identify the correlation you would expect to find. Explain. 1. the number of knots tied in a rope and the distance between the rope s ends. Each knot decreases the distance between the rope s ends. As the number of knots increases, the distance between the ends decreases. The correlation is negative.. the number of grams of fat and the number of calories in cheese positive correlation; As the amount of fat in a food increases, the calories increase. 3. the height of a student and his or her score on an algebra test no correlation; There is no relationship between height and algebra skill. Saxon. All rights reserved. 155 Saxon Algebra 1

4 continued 71 By finding the equation of a trend line drawn over a graph of data, you can make predictions. The scatter plot shows a relationship between a man s height and the length of his femur (thigh bone). Based on this relationship, predict the length of a man s femur if his height is 1 centimeters. Step 1: Draw a trend line through the points. Step : Locate 1 centimeters on the x-axis. Step 3: Go up to the line. Step : Go left from the line to the y-axis. The point (1, 1) is on the line. Your line should have about as many points above as below. It may or may not pass through plotted points. Femur length (cm) y 3 x A man who is 1 centimeters tall would have a femur about 1 centimeters long. Height (cm). Predict the height of a man who has a femur that is centimeters long. Use the scatter plot shown above. The point (1, ) is on the line. A man with a cm femur would be about 1 cm tall. The scatter plot shows the relationship between engine size and fuel economy for ten automobiles. 5. Draw a trend line on the graph.. Use the trend line to make the following predictions. a. the city fuel economy of an automobile with an engine size of 5 L Sample: 8 mi/gal b. the city fuel economy of an automobile with an engine size of.8 L Sample: mi/gal c. the engine size of an automobile with a city fuel economy of 11 miles per gallon Sample:.5 L Fuel Economy (mi/gal) y Engine Size (L) x Saxon. All rights reserved. 15 Saxon Algebra 1

5 Name Date Class You have multiplied two binomials. Now you will reverse this process to factor a trinomial into two binomials. When factoring : x bx c: c is the product of the last terms of the binomials. b is the sum of the last terms of the binomials. If c is positive and b is positive and b is negative Factoring Trinomials: x bx c 7 both factors of c are positive. both factors of c are negative. Factor x 7x 1. Need factors of 1 that sum to 7. Factor of 1 Sum 1 and 1 11 and 5 1 x 7x 1 (x )(x 5) Factor x 9x 18. Need factors of 18 that sum to 9. Factor of 18 Sum 1 and and and 9 x 9x 18 (x 3)(x ) Complete the steps to factor each trinomial. 1. x 1x 1. x 11x Need factors of 1 that sum to 1. Need factors of that sum to 11. Factor of 1 Sum Factor of Sum 1 and and 5 and 8 and 1 1 and and 8 11 and 1 x 1x 1 (x )(x 8) x 11x (x 3)(x 8) Factor each trinomial. 3. x 13x 1. x 15x 5 (x 1)(x 1) (x 5)(x 1) 5. x 13x 3. x 11x 3 (x )(x 9) (x 5)(x ) 7. x 1x 1 8. x 1x 3 (x 7)(x 3) (x )(x 8) Saxon. All rights reserved. 157 Saxon Algebra 1

6 continued 7 If c is negative and b is positive and b is negative the larger factor of c must be positive. the larger factor of c must be negative. Factor x 8x. Then evaluate x 8x and its factors for x 3. Need factors of that sum to 8. Factor x 3x 8. Then evaluate x 3x 8 and its factors for x 3. Need factors of 8 that sum to 3. Factors of Sum Factors of 8 Sum 1 and 19 1 and 8 7 and 1 8 and 1 1 and 5 1 x 8x (x )(x 1) Evaluate x 8x and its factors for x 3. Trinomial Factors x 8x (x )(x 1) (3) 8(3) (3 )(3 1) 9 (1)(13) The results are the same. and 7 3 x 3x 8 (x )(x 7) Evaluate x 3x 8 and its factors for x 3. Trinomial Factors x 3x 8 (x )(x 7) (3) 3(3) 8 (3 )(3 7) (7)( ) 8 8 The results are the same. Complete the steps to factor each trinomial. 9. x x 1. x 3x Need factors of that sum to 1. Need factors of that sum to 3. Factors of Sum 1 and 19 and 1 8 and 5 1 x x (x )(x 5 ) x 3x (x 1 )(x ) Factor each trinomial. 11. x 3x 18 (x 3)(x ) 1. x 5x 1 (x )(x 7) 13. Evaluate x x 3 and its factors for x 5. Factors of Sum 1 and 3 and x x 3 (x 1)(x 3); 5 (5) 3 1; (5 1)(5 3) 1 Saxon. All rights reserved. 158 Saxon Algebra 1

7 Name Date Class Solving Compound Inequalities 73 You have solved inequalities and graphed their solutions. Now you will solve and graph compound inequalities. Conjunctions are compound inequalities using AND. They require you to find solutions so that two inequalities will be satisfied at the same time. Solve the conjunction x 3 5 and graph the solution. Solve x 3. x 3 Solve x 3 5. x 3 5 _ 3 3 _ Add 3 to both sides. _ 3 3 _ 1 x x Graph x 1. Graph x. Graph 1 x Add 3 to both sides. Use overlapping regions for compound inequalities with AND. Complete the steps to solve the conjunction and graph the solution x 8 One inequality is 3 x. The other inequality is x 8. 3 x x 8 Add to both sides. Add to both sides. 1 x x 1 Graph 1 x Solve the conjunction and graph the solution.. 8 m 15 m k k 1 Saxon. All rights reserved. 159 Saxon Algebra 1

8 continued 73 Disjunctions are compound inequalities using OR. They require you to find solutions that satisfy either of the two inequalities. Solve the disjunction x 1 OR 3x 15 and graph the solution. Solve x 1. Solve 3x 15. x 1 3x 15 x 1 Divide both sides by. 3x 15 Divide both sides by x 3 x 5 Graph x 3. Graph x Use both regions for compound inequalities with OR. Graph x 3 OR x Complete the steps to solve the disjunction and graph the solution.. x 1 OR x Solve x 1. Solve x. x 1. x. Add to both sides. x 3 x 3 OR x. Solve the disjunction and graph the solution. 5. x 3 5 OR x - - x x Divide both sides by. x OR x b OR 3b 3-8 b 7 OR b 1 Write a compound inequality that describes each graph x 3 OR x x 1 Saxon. All rights reserved. 1 Saxon Algebra 1

9 Name Date Class You have used absolute values to add real numbers. Now you will solve absolute-value equations. Absolute-value equations are equations that have one or more absolute-value expressions. Write an absolute-value equation as two equations to find two solutions. Solve x 15. Solve x 15. x 15 _ _ Add to each side. x 17 The solution is { 13, 17}. Solve x 15. x 15 _ x 13 It is possible for an absolute-value equation to have only one solution. Solve x 3. Solve x 3. Solve x 3. x 3 _ 3 x 3 The solution is { 3}. _ 3 Add 3 to each side. Complete the steps to solve the absolute-value equation. 1. x 1 Solving Absolute-Value Equations 7 _ Add to each side. This is the same equation as x 3. Solve x 1. x 1 Solve x 1. x 1 Add to each side. Add to each side. x x 1 The solution is { 1 }. Solve the absolute-value equations.. y 9 3. b 5 13 { 9 9} { 18 8}. c 1 5. m 3 {1} {1 7} Saxon. All rights reserved. 11 Saxon Algebra 1

10 Sometimes it is necessary to isolate the absolute-value term on one side of the equation before solving. Solve 3 x. 3 x 3 3 x Divide both sides by 3. continued 7 Solve x. x _ _ Add to each side. x The solution is {, }. Solve x. x _ _ Add to each side. x Complete the steps to solve the absolute-value equation. x x 5 1 Multiply both sides by 5. 5 x 5 Solve x 5. x 5 Solve x 5. x 5 Add to both sides. Add to both sides. x 7 The solution is { 3, 7}. x 3 Solve the absolute-value equations. 7. t 11 {11} 8. 7 x 1 1 {, } 9. k 5 1 {1, 9} 1. x 3 7 { 17, 11} 11. A yogurt plant packages fat-free yogurt in 8-ounce containers. A quality control engineer inspects the weight of a batch of filled containers. He rejects any container whose weight varies from 8 ounces by.5 ounce. Write and solve an absolute-value equation to find the minimum and maximum weights that the engineer will accept. x 8.5, {7.75, 8.5} Saxon. All rights reserved. 1 Saxon Algebra 1

11 Name Date Class Factoring Trinomials: ax bx c 75 You have factored trinomials of the form x bx c. Now you will factor trinomials of the form ax bx c. When factoring ax bx c, first find factors of a and c. Then check the products of the inner and outer terms to see if the sum is b. Factor x 11x 15. x 11x 15 ( x )( x ) Factors of Factors of 15 Outer Inner 1 and 1 and and 15 and and 5 and and 3 and x 11x 15 (x 3)(x 5) Factor 3x 3x 1. 3 x 3x 1 ( x )( x ) Factors of 3 Factors of 1 Outer Inner 1 and 3 1 and 1 1 ( 1) 3 ( 1) 17 1 and 3 1 and 1 1 ( 1) 3 ( 1) 3 1 and 3 and 7 1 ( 7) 3 ( ) 13 1 and 3 7 and 1 ( ) 3 ( 7) 3 3x 3x 1 (x 7)(3x ) Complete the steps to factor the trinomial. 1. 5x 1x Factors of 5 Factors of Outer Inner 1 and 5 1 and and 5 and and 5 and x 1x (x )(5x ) Factor each trinomial.. 3x 7x (3x )(x 1) 3. x 13x 1 (x 7)(x 3). x 8x 3 (x 3)(x 1) 5. x 9x 1 (x 5)(x ) Saxon. All rights reserved. 13 Saxon Algebra 1

12 continued 75 When c is negative, one factor of c is positive and one is negative. You can stop checking factors when you find the factors that work. Factor x 7x 15. x 7x 15 ( x )( x ) Factors of Factors of 15 Outer Inner 1 and 3 and ( 3) 1 1 and 3 and 5 1 ( 5) and 5 and ( 5) 7 1 and 5 and 3 1 ( 3) 5 7 x 7x 15 (x 5)(x 3) When a is negative, factor out 1. Then factor as shown previously. Factor 5x 8x 1. 1( 5x 8x 1) 1 ( x )( x ) Factors of 5 Factors of 1 Outer Inner 1 and 5 and 1 5 ( ) 1 and 5 and 1 ( ) 5 1 and 5 and 1 ( ) and 5 and 1 5 ( ) 8 5x 8x 1 1(x )(5x ) Complete the steps to factor the trinomial.. 3x 7x. 3x 7x 1(3 x 7x ) Factors of 3 Factors of Outer Inner 1 and 3 and 5 1 ( 5) 3 () 7 1 and 3 and ( ) 7 3 x 7x 1(x )(3x 5) Factor each trinomial x 3x 7 8. x 3x 5 (5x 1)(x 7) 1(x 5)(x 1) 9. 3x 7x 1. x x 15 (3x )(x 3) (x 5)(x 3) Saxon. All rights reserved. 1 Saxon Algebra 1

13 Name Date Class You have simplified expressions that contain radicals. Now you will multiply binomials containing radicals and simplify them using the Product of Radicals Rule. Product of Radicals Rule The square root of a product equals the product of the square roots of the factors. ab a b where a and b Example: (3)() is the same as (3) Simplify: 9. Use the Product of Radicals Rule. 9 ()(9) a b ab where a and b 3 Multiply: 9 3. Simplify: 3. Simplify: 5. Use the Product of Radicals Rule. 5 (5) Square each factors separately. 5 1 Simplify: 5 5; 1. 5 Simplify: 1. 1 Multiply. Multiplying Radical Expressions 7 Complete the steps to simplify each expression ( )( 5 ) 3 1 ( 3 ) Simplify. All variables represent non-negative real numbers x 9x 7. x 3x x Saxon. All rights reserved. 15 Saxon Algebra 1

14 continued 7 Remember to either use the square of a binomial pattern, the Distributive Property, or the FOIL method when multiplying polynomials. Simplify: 3 8. Use the Distributive Property Multiply: Simplify. Simplify: 3. Use FOIL to multiply binomials with square roots. 3 3() Factor using a perfect square factor. Use the Product of Radicals Rule. FOIL. Multiply. Simplify. Add. Complete the steps to simplify each expression Simplify (5) (8 1 ) Mr. Hoy has a square garage. The garage s side length is 8. What is the area of the garage floor? 1 square units Saxon. All rights reserved. 1 Saxon Algebra 1

15 Name Date Class You have solved one-step inequalities. Now you will solve inequalities that require more than one step to solve. Solve the inequality 5x 3 3 and graph the solution. Two-step and multi-step inequalities require more than one inverse operation to isolate the variable. Step 1: Isolate the variable on one side of the inequality sign. 5x 3 3 _ 3 3 _ 5x Subtract 3 from both sides. 5x Divide both sides by 5, and 5 5 x reverse the direction of the inequality sign. Step : Graph the solution. Place an open circle at and shade to the right Solving Two-Step and Multi-Step Inequalities 77 Complete the steps to solve each inequality and graph the solutions. 1. 3m 1 3m 1 _ 1 _ 1 3m 3m 3 3 m s 15 3 s _ 3 _ 3 1 s 1 s 3 s Solve each inequality and graph the solutions x 1 x y 3 1 y c 8 11 c j 1 j Saxon. All rights reserved. 17 Saxon Algebra 1

16 Some inequalities may require using the order of operations, the Distributive Property, combining like terms, or multiplying by the LCM of the denominators. Solve the inequality 3 x 1 5 and graph the solution. 3 Two-step and multi-step inequalities require more than one inverse operation to isolate the variable. Step 1: Multiply by the LCM of the denominators, then solve. 3 x () x 1() 5() 3 9x x 9 x 1 Step : Graph the solution. Place a closed circle at 1 and shade to the right continued 77 Complete the steps to solve each inequality and graph the solutions x 3 1 5( ) x 3( 5 x 3 1 ) 1( ) 5x x 15 x (g ) g 1 (g ) g g 8 g 1 ( 3 ) 3 g 8( 3 ) 3 3 g( 3 ) 1( 3 ) g 8 3g 3 g 5 g 5 Simplify k 3k 7 5 k (b 7) b 1 b Saxon. All rights reserved. 18 Saxon Algebra 1

17 Name Date Class You have simplified rational expressions. Now you will graph rational functions. Find the excluded values of y t t 8. Because rational functions have variables in the denominator, any value that makes the denominator equal to zero must be excluded. Rational functions are undefined when the denominator is equal to zero. Set the denominator equal to zero and solve for t. t 8 _ 8 _ 8 t 8 The denominator is t 8. Set it equal to zero. Subtract 8 from both sides. Simplify. 8 is an excluded value, because when t 8, the denominator is zero so the function is undefined. Complete the steps to find the excluded values. Graphing Rational Functions y 7 3x 3x x. y x 5 x 5 x 5 For the function, is an excluded value. This is because when x the function is undefined. Find the excluded values. 3. y x 7 5. y 7 3 x y x 9. y 1 3x 1 3x 3 For the function, 5 is an excluded value. This is because when x 5 the function is undefined.. y. y 3 3 x x 1 x 3 8. y x 1 x y x 1 3x 9 3 Saxon. All rights reserved. 19 Saxon Algebra 1

18 continued 78 Most rational functions are discontinuous functions. Discontinuous functions are functions that have a break, hole, or jump in the graph. A break or jump in the graph can be due to an asymptote. A vertical asymptote is a boundary line that the graph of the function approaches but never touches or crosses. A horizontal asymptote guides the end behavior of the function. Identify the asymptotes and graph the function y 1 x. Step 1: Identify the horizontal and vertical asymptotes. y 1 x Rational function in the form y y 1 a 1, b, and c. x ( ) Since b, the equation for the vertical asymptote is x. Since c, the equation for the horizontal asymptote is y. Step : Graph the asymptotes using dashed lines. Step 3: Make a table of values. Choose x values on both sides of the vertical asymptote. a c where x b (-8,-) (-,-3) (-,-) 1 8 O - -8 y 1 y = x+ (1,) (,3) (,) 8 1 x x 8 1 y Complete the steps to identify the asymptotes and graph the function. 11. y x 1 x 1 x 1 The equation for the vertical asymptote is x 1. Since c, the equation for the horizontal asymptote is y. x y 1 1 O (-3, -1) - (-1, -) (, -) - y (,) (3,) (5,1) x Identify the asymptotes and graph the function. 1. y x ; y 13. y 3 x x y 1 x ; y 1 y O - x O - x - - Saxon. All rights reserved. 17 Saxon Algebra 1

19 Name Date Class Factoring Trinomials by Using the GCF 79 You have simplified algebraic expressions by using the greatest common factor, or GCF. Now you will factor polynomials by using the GCF. To factor a polynomial completely, first find the GCF of the terms and then factor it out. Factor x 5x 3 1x completely. Step 1: Find the GCF of all the terms in the polynomial. Prime factorization of x is: x x x x. Prime factorization of 5x 3 is: 1 5 x x x. The GCF is x. Prime factorization of 1x is: 1 7 x x. Step : Write terms as products using the GCF. x 5x 3 1x ( x ) x ( x )5x ( x )1 Step 3: Use the Distributive Property to factor out the GCF. x ( x 5x 1) Step : Factor x 5x 1. Find two numbers that have a product of 1 and a sum of and 7 5. x 5x 1 (x 7)(x ) x 5x 3 1x factored completely is x (x 7)(x ). Complete the steps to factor each trinomial completely. 1. x 3x 3 1x. x 3 x 1x x 3x 3 1x x ( x 3x 1) x ( x 5)( x ) x (x 5)(x ) x 3 x 1x x( x x ) x( x 3)(x ) x (x 3)(x ) Factor completely. 3. x 1x 3 3x x ( x )(x 8) 5. x 3 1x x x(x 8)(x 8) 7. x 3 1x 8x x(x 7)(x ). x x 3 x x (x 3)(x ). 3x 3 x 8x x(3x )(x ) 8. 3x x 3 3x 3x (x 1)(x 1) Saxon. All rights reserved. 171 Saxon Algebra 1

20 continued 79 When the leading coefficient of a trinomial is negative, factor out a 1. first. Factor x x 7 completely. Step 1: Factor out 1. x x 7 1( x x 7) Step : Factor x x 7. Find two numbers that have a product of 7 and a sum of 1. ( x x 7) (x 8)(x 9) and 8 ( 9) 1 x x 7 factored completely is (x 8)(x 9). Complete the steps to factor each trinomial completely. 9. x x 1. x 3 3x 18x x x 1( x x ) Factor completely. 11. x x 1 (x )(x 3) 13. x x 8 (x )(x ) 1(x 7)(x ) (x 7)(x ) x 3 3x 18x x( x 1x ) 1. x x 15 (x 3)(x 5) 1. x 3x 7 (x 7)(x 1) x(x 8)(x 8) x(x 8)(x 8) 15. x 17x 3 (x 15)(x ) 17. x x (x 5)(x ) 19. 3x 3 x x 3x(x 1)(x ) 1. x x 8 (x 8)(x 1) 18. x 3 x 7x x(x 5)(x 7). 5x 3 1x 5x 5x(x 1)(x 1) 1. The height of a golf ball hit from an elevated tee 35 feet above the hole is 5t 3t 35, where t represents time. Factor the expression completely. 5(t 1)(t 7) Saxon. All rights reserved. 17 Saxon Algebra 1

21 Name Date Class Now you will learn how to use tables and graphs to represent probability. For lunch, Omar can choose a sandwich made of ham, tuna, or cheese. He can also add either lettuce or tomato. Make a table to show all his possible sandwich choices. Use the sandwich choices for the column headings. Use the add-ons for the row headings. Record all the possible choices. Ham Tuna Cheese Lettuce HL TL CL Tomato HT TT CT Calculating Frequency Distributions 8 Find the probability of each outcome as a fraction in lowest terms. There are different sandwich possibilities. Use as the denominator. Each possibility occurs only once. Use 1 as the numerator. The probabilities for each type of sandwich are: P(HL) 1 P(HT) 1 P(TL) 1 P(TT) 1 P(CL) 1 P(CT) 1 Complete the table to show the possible outcomes. Then find the probability of each outcome. 1. There are blue and 3 white marbles in a bag. There are 1 possible outcomes. Kim chooses a marble, then flips a coin. BH appears times. P(BH) BT appears times. P(BT) Blue Blue White White White Heads BH BH WH WH WH Tails BT BT WT WT WT Complete the table to show the possible outcomes. Then find the probability of each outcome.. There are 3 pennies, nickels, and 1 dime in Shauna s pocket. She takes one coin at a time out of her pocket and flips it WH appears 3 times. P(WH) 3 1 WT appears 3 times. P(WT) 3 1 Penny Penny Penny Nickel Nickel Dime Heads PH PH PH NH NH DH Tails PT PT PT NT NT DT P(PH) P(PT) 1 1 P(NH) P(NT) 1 1 P(DH) P(DT) Saxon. All rights reserved. 173 Saxon Algebra 1

22 continued 8 You can represent the frequency distribution of data with a graph. Anna kept track of the color of all the vehicles that passed on her morning walk. The table shows Anna s data. Make a bar graph to show the frequency distribution of the data. 7 Color Frequency Silver 3 White Red 5 Frequency Silver White Red Find the probability for each vehicle color. Express each probability as a fraction in lowest terms. P(silver) 3 1 Colors P(white) P(red) 5 1 Make a bar graph that shows the frequency distribution of the data in the table. Then find the probability of each outcome as a fraction in lowest terms. 3. Evan keeps track of the birds he sees on his hikes. Bird Frequency Blue jay 7 Hawk 1 Sparrow 5 Frequency P(B) 7 13, P(H) 1 13, P(S) 5 13 Blue jay Hawk Sparrow. Maria asked 1 students how many books they were taking home over the weekend. She placed the data in a table. Number of Books Frequency 5 3 Frequency P() 1, P(3) 1, P() Number of books Saxon. All rights reserved. 17 Saxon Algebra 1