1.1 Real Numbers and Number Operations

Transcription

1 1.1 Real Numbers and Number Operations The Real Number System Create a Venn Diagram that represents the real number system, be sure to include the following types of numbers: real, irrational, rational, integers, wholes, and natural/counting. Properties of Real Numbers Let a, b and c be real numbers. Property Addition Multiplication Commutative a + b = b + a a b = b a Associative (a + b) + c = a + (b + c) (ab) c = a (bc) Identity a + 0 = a ; 0 + a = a a 1 = a, 1 a = a Inverse a + (- a) = 0 a 1 = 1 ; (a 0) a Distributive Property a(b + c) = ab + ac Identify the property illustrated. a.) = b.) 5 1 = 1 c.) (5 + 3) + 2 = 5 + (3 + 2) 5 d.) 5 ( x + 2) = 5x + 10 e.) = 0 f.) 1 5 = 5

2 Unit Analysis Conversion Reference Table English Units Metric Units English / Metric Units 1 ft = 12 in 1 m = 100 cm 1 in = 2.54 cm 1 yd = 3 ft 1 cm = 10 mm 1 mi = 1.61 km 1 mi = 5280 ft 1 kg = 1000 g 10 km = 6.2 mi 1 lb = 16 oz 1 L = 1000 ml lb = 454 g 1 gal = 4 qt 1 L = qt Try the following conversions. 1.) 2.45 mi = ft 2.) 36 g = kg 3.) 470 mi = km 4.) 200 mi/hr = ft/sec 5.) 526 yds/sec = mi/hr 6.) 75.0 kg = lbs. 7.) 10 gal = ml 8.) 1.43 kg/l = g/ml

3 1.1 Real Numbers and Number Operations Homework 1. Give an example of a number that is both a rational number and a whole number. 2. Give an example of a number that is both an integer and a natural number (counting number). 3. Give an example of a number that is a rational number, but NOT an integer. 4. Name the property illustrated in each example below: a) 5(x + 7) = 5x + 35 b) (59 + 7) + 62 = 59 + (7 + 62) c) 10 i 1 10 = 1 d) = e) = 387 f) 4(3) = 3(4) g) 0 = 5 + (- 5) 5. Perform the following conversions using conversion factors. Show your set up. 86 inches = ft 5.17 lb/gal = lb/qt 2.4 g/ml = lb/gal 3.4 km/hr = mi/hr

4 Use what you have learned about conversion factors to solve the following application problems. 6. The elevator in the Washington Monument takes 75 seconds to travel 500 ft to the top floor. What is the speed of the elevator in miles per hour? 7. When you drive across the border into Quebec, Canada, all the speed limit signs are in km/hour. How might you get a quick, reasonable approximation of the equivalent speed in mi/hour? 8. As a nurse, you must determine the proper dose of a medication for your patient who weighs 160 pounds. The amount of medication is 20 mg/kg. (mg of medication per the patient s weight in kilograms) How many milligrams of medication would be a proper dose for your patient? 9. A blood donor gives 1 pint of blood each time he donates. He learns from the American Red Cross that he will be honored at a special dinner for outstanding donors for his lifetime donation of 6 gallons of blood. How many times has he donated blood? (There are 2 pints per 1 quart of any liquid.)

5 1.2 Algebraic Expressions and Models & 1.3 Solving Linear Equations Order of Operations - PEMDAS 1.) Undo grouping symbols parentheses, absolute value, braces, brackets 2.) Evaluate any exponents / powers 3.) Do multiplication and division as they appear left to right 4.) Do addition and subtraction as they appear left to right Evaluate (1 (- 3)) 3 2. ( 3) Evaluate 1. 3x 2 5x + 7 when x = x 3 + 3x when x = 4 Simplifying Algebraic Expressions 1. 5n 2 (n +1) 3n 3 2n 2. 2[(3n +1) 2 n] Solving Equations 1. 5(x 2) = 4(2x + 7) + x x = x x = 1 5x 1 5 ( ) x = x + 1 5

6 5. 6x + 2 4x = 3(2x + 1) 2 2x (2x + 3) = 2(4 3x) 4x Word Problem Practice 1.) Find four consecutive even integers that total ) Find three consecutive integers such that if the sum of the first two is decreased by the third, the result will be ) Find three consecutive even integers such that twice the smallest plus three times the largest will equal the middle integer increased by ) Mrs. Bunker paid her $7.15 grocery bill with a ten- dollar bill and asked for her change in dimes and quarters. If she received 21 coins in change, how many quarters did she receive? 5.) 65 students and teachers from Penncrest High School are going to a play. Tickets costing $4 for teachers and $3.60 for students were purchased at a total price of $244. How many $4 tickets were bought? 6.) A stamp collector has a group of ten cent, fifteen cent and twenty cent stamps that total $4.75 in value. If he has twice as many twenty- cent stamps as ten cent stamps and three fewer fifteen cent stamps than 10 cent stamps, how many twenty cent stamps does he have?

7 1.2 and 1.3 Homework Evaluate the following expressions. 1.) (3) 4 2.) ( 2) 4 3.) ) ( 2) 4 5.) (5 2) ) ((3 1) 2 + ( 3)) 5 7.) 1 2 ( 2) 2 + (4 5) 4 8.) x 2 4xy when x = 2 and y = 3 9.) 6 x 2 + x when x = 2 Simplify the following expressions. 10.) 5(n 2 + n) 3(n 2 2n) 11.) 8(y x) 2(x y) Solve the following equations. 12.) 6(2x 1) + 3 = 6(2 x) 1 13.) 5x + 2 = 2(2x+1) + x 14.) 5(2x + 3) = 2(4 3x) 4x 15.) 1 2 x 5 3 = 1 2 x

8 16.) x 5 3 = 2 17.) x 7 10 = ) Find three consecutive integers such that if three times the smaller is decreased by the sum of the other two, the difference will be ) Nancy has a bag of coins totaling $3.60 in value. If she has two more nickels than quarters, and twice as many quarters as dimes, how many of each coin does she have?

9 1.4 Rewriting Equations and Formulas Warm up: Solve the equation 3 2 x = 7 2 x Example 1: Solve for y: 11x 9y = 4 Example 2: Solve for y: 6x y = 21 Example 3: Solve for y. xy + 2x = 20

10 Solve each formula for the indicated or underlined variable. a.) P = 2l + 2w b.) A 2 = π r c.) 1 A = bh (solve for h) 2 d.) E 2 = mc (solve for m) e.) 9 F = C + 32 f.) S = C rc (solve for C) 5 g.) F = Gm 1m 2 r 2 (for m2) h.) I = E R + r (for R) i.) A = 1 2 h ( b + b 1 2 ) for b2 j.) 1 R = (for R2) R 1 R 2 k.) S = 2WH + 2WL + 2LH (for H)

11 1.4 Homework: Rewriting Equations and Formulas 1.) A = P + Prt (solve for t ) 2.) PV = nrt (solve for R) 3.) P 1 V 1 T 1 = P 2V 2 T 2 (Solve for P 2 ) 4.) S = V 0 t 16t 2 ( Solve for V 0 ) 5.) V = 1 3 πr2 h (Solve for h) 6.) 1 f = 1 a + 1 b (Solve for b) 7.) P = R C n (Solve for R) 8.) S = 2πr 2 + 2πrH (Solve for H)

12 1.4 Homework Day 2

13 1.6 Solving Linear Inequalities Graph the following inequalities: - 4 < x 5 x And inequalities x - 3 and x 4 x < 3 and x 6 Or inequalities x 4 or x 0 x > 10 or x 6 Solve and Graph: 1.) - 11y ) 2x + 1 6x 1 3.) 12 < 3x 3 <15 4.) 2x or 3x + 5 2

14 1.6 Homework Match the inequality with its graph. 1.) x 4 2.) x < 4 3.) 4 < x 4 4.) x 4 or x < ) - 4 x 4 6.) x > 4 or x - 4 Determine whether the given number is a solution to the inequality. 7.) 1 x 2 4 Is 9 a solution to this inequality? 3 8.) 8 < x 11 < 6 Is 5 a solution to this inequality? Solve the inequality. Then graph your solution. 9.) x 6 10.) 5 5x > 4(3 x)

15 11.) 5 n ) 8 < 2 3 x 4 < ) 3x + 2 < 10 or 2x 4 > 4 14.) 3n + 1 > 10 and 1 2 n 1 > 3 15.) 2(x + 3) < 14 or 5 x < 1

16 1.7 Absolute Value Equations and Inequalities General Form: ax + b = c The solution set for this equation can be found by solving these two equations: Example: 5x + 2 = 27 ax + b = c ax + b = c General Form: ax + b > c The solution set for this general form is the union of two sets, namely, ax + b > c or ax + b < c When you solve inequalities of this form, rewrite the original inequality as the union of two separate inequalities. Solve and graph the union of the solution sets. Write the solution set using set notation. Example: 3x + 4 5

17 General Form: ax + b < c The solution set for this general form is the intersection of two sets, namely, which can be written more simply as c < ax + b and ax + b < c c < ax + b < c When you solve inequalities of this form, rewrite the original inequality as the union of two separate inequalities. Solve and graph the intersection of the solution sets. Write the solution set using set notation. Example: 4x Practice Problems: 1 1.) x 3 = 2 2.) 2 x + 7 < 11 `2 3.) 3x +10 > 7 4.) x + 5 6

18 5.) 2x ) 4 2y < 9 7.) 3x 5 < 1 8.) 3x 5 > 1 9.) 2x ) x 3 > 0