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2.1 The Real Number Line Objective: To be able to graph numbers on the number line and to find the opposite and absolute value of numbers. Negative Numbers Positive Numbers -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 Origin Graph and label 5.5, ½, –4.25, –9/4 Positive numbers: Points to the right of zero Ex: 4.3 Negative numbers: Points to the left of zero Ex: -1.6 Zero: Neither positive nor negative Ex: 0 Integers: Negative and positive numbers & zero Abbr: Z Ex:-6, 4, 0 Real numbers: All integers and all numbers in Abbr: R Ex:-2.8, -4, between (fractions and decimals) 0, 7, 8/3 II. Comparing Real Numbers • We compare numbers in order by their location on the number line. • Graph –4 and –5 on the number line. Then write two inequalities that compare the two numbers. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 Since –5 is farther left, we say –4 > –5 or –5 < –4 • Put –1, 4, –2, 1.5 in increasing order -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 Left to right –2, –1, 1.5, 4 II. Comparing Real Numbers • Write the following set of numbers in increasing order: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 –2.3, –4.8, 6.1, 3.5, –2.15, 0.25, 6.02 –4.8, –2.3, –2.15, 0.25, 3.5, 6.02, 6.1 III. Fraction Time!!! • To convert a fraction to a decimal, Divide the numerator by the denominator • Write each set of numbers in increasing order. a. 2 , 0.3, 5 , 0, 0.25, 2 b. 5 1 , 3.8, 9 , 4.8, 6 3 2 9 2 2 5 5 2 2 6 9 1 , 0, , 0.25, 0.3, , 3.8, , , 4.8, 5 2 9 3 5 2 2 • YOU TRY a, c, and d! 1 2 5 8 7 c. –3, -3.2, -3.15, -3.001, 3 , d. 3 7 , , , 3 3 9 3.2, 3.15, 3.001, 3, 3 8 5 2 1 7 , , , , 3 3 7 3 9 IV. Opposites • Opposites: Two points (numbers) that are the same distance from the origin but on opposite sides of the origin 8 units away 8 units away from origin from origin -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 • Name the opposite of the following numbers: a. 10 b. -½ c. 4.5 d. 0 –10 1/2 –4.5 0 V. Absolute Value • What is absolute value? – The distance a number is from origin . – Distance is ALWAYS Non-negative ! • What does it look like? We say “the absolute value of x” Think: “How many x Think: “How many units away from 5 units away from -5 Absolute to zero? to zero? value bars 5 5 5 5 Find the absolute value of the following: 4 4 6 6 0 0 4 4 3 3 • Tricky Case: How are these different from each other? From the ones on the previous slide? 4 –4 4 –4 VI. Solving Absolute Value Equations Using Mental Math • Ex. x 3 x 3 or –3 Think, “What number(s) So… How many is/are 3 units • Ex. x 7 x 7 or –7 answers can away from these have? zero? • Ex. x 2 x No solution • Ex. x 0 x 0 Don’t confuse that “No solution” and “x = 0” are different!!! Summary 1. Graph the real numbers on the number line. 2. Compare the real numbers (rewrite the real numbers in increasing/decreasing order). The number at the LEFT is smaller than the one at the RIGHT. 3. When converting a fraction to a decimal, just divide the numerator by the denominator. 4. Opposites are the two real numbers “mirror” each other. 5. Absolute value is the distance a number on the number line from the origin. Can you compare and order real numbers and find the opposite and absolute value of numbers?? 2.2-2.3 Adding and Subtracting Real Numbers Objective: To add and subtract real numbers and to review rules for addition and subtraction of fractions and decimals. Please, no calculators today Let‟s see if we all get the same answer for 4 + 3 – 2 + (-5) + 7 – 1 + 3 + (-4) - 5= _____ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 A few rules: Properties of Addition Definition Example Commutative a+b=b+a 4 + (-2) = (-2) + 4 Associative (a + b) +c=a +(b + c) [(-4) + 3] + (-2) = (-4) + [3 + (-2)] Identity a+0=a (-2) + 0 = (-2) Inverse a + (-a) = 0 7 + (-7) = 0 Procedures for Adding Real Numbers • Adding Same Signs: • Ex: -5 + -6 = Ex: 5 + 6 = • Rule: Signs alike Add and keep the sign Same Signs Do “Sum” (Add) and keep the sign • Adding Different Signs: • Ex: -4 + 9 = Ex: 4 + (-9) = • Rule: Different Signs Do “Diff ” (Subtract) and keep the sign of what you have more of ASK!! Decimals and Fractions • Decimals: Ex: 4.03 + 3.142 = Ex: -2.15 + 4.2 Line up the decimals and fill in 0s to make the numbers Rule: have the same length. Be careful with subtract – The number with the larger absolute value must go first!! Fractions: 2 1 4 3 Ex) 1 Ex) 9 3 3 4 Rule: Make common denominators Yes I need to see your work. You add/subtract the numerators, but keep the denominators the same. If you choose the „least common denominator‟ you will do less work. So… What do we do with subtraction? Ex 1) 12 3 Ex 2) 6 ( 4) 1 4 Ex 3) 18.3 ( 4.514) Ex 4) 5 15 • Rule: 1) Change subtraction to „add the opposite‟. then follow the rules of addition. 2) Directly subtract. **Cross the line change the sign.** **Make a “Smiling” face.** Now you try Ex 1) 14 3 Ex 2) 8 ( 8) 2 5 Ex 3) 23.3 ( 4.514) Ex 4) 3 12 • Rule: 1) Change subtraction to „add the opposite‟. then follow the rules of addition. 2) Directly subtract. **Cross the line change the sign.** **Make a “Smiling” face.** Evaluate the function when x = –2, –1, 0, 1, 2 and y = x –6 x y –2 –8 y = –2 – 6 = –2 + (–6) = –8 –2 –7 y = –1 – 6 = –1 + (–6) = –7 0 –6 y = 0 – 6 = 0 + (–6) = –6 1 –5 y = 1 – 6 = 1 + (–6) = –5 2 –4 y = 2 – 6 = 2 + (–6) = –4 Summary Adding Fractions Decimals Whole Numbers Get Common Line Up the Denominators! Decimals Same Sign Opposite Signs Subtract Add Keep the sign of what you have Keep the sign more of Subtracting Change to Cross the Line, Two Negative Addition Change the Sign signs Make a Positive Then Follow Your Addition Rules! 2.4 Adding Matrices Objective: To add/subtract matrices Vocabulary: What is a matrix?? matrix (matrices) – rectangular arrangement of numbers into horizontal rows and vertical columns Entry or element – a number in a matrix ex: 3, ½ , 0, 8, -1, 2, 4, -2, 5, size of a matrix – (# rows) x (# columns) (really important!!) ex: 2 x 3, 3 x 3 4 6 1 6 2 1 A B 2 0 8 2 0 5 3rows 2 rows 9 3 12 3 columns 3 columns equal matrices – matrices with equal entries in corresponding positions What does this mean?? 2 5 Ex) 6 1 2 5 x y Ex 1) w x = 2, y = –5 , w = 6, k = 1 6 1 k 4 c b 12 Ex 2) H 3 a = 3, b = –4 , c = 12, d = 5 a 5 d 4 12 What is H = 3 5 **State law prohibits you from adding or subtracting matrices of different sizes.** Procedure: Add corresponding entries 2 3 5 1 7 4 Ex: 1 4 12 2 11 6 0 2 3 16 3 14 3x2 3x2 3x2 Ex. Find the values for all variables. 12 6 4 7 w x 4 1 3 2 y z w = 8, x = 1, y = –7 , z = 1 You try these!!! Remember - lk to make sure the matrices are the same size before you try to add or subtract!!! Otherwise…. What do you do?? Ex1) 1 5 3 4 10 2 5 -15 5 10 6 4 5 14 1 Ex2) 3 2 2 5 0 2 4 6 5 6 5 Ex3) 1 1 No solution 3 0 1 7 9 Give the dimensions of each matrix. So… What do we do with subtraction of matrices? 1 7 6 3 4 12 1 4 Ex1) 5 2 3 Ex2) No 4 5 1 0 7 solution 2 0 2 1 2 7 1 8 3 Ex3) 10 7 6 5 4 12 Add/subtract the numbers/entries in the corresponding Rule: positions. Summary 1. The size of the matrix is described by row number x column number. 2. Equal matrices are the matrices with equal entries in corresponding positions. 3. Adding/subtracting matrices of the same size is to add/subtract the entries in corresponding positions. 4. It can add/subtract if two matrices of different size. Assignment p. 89 Q12 - 25 all 2.5 Multiplying Real Numbers Objective: To multiply positives and negatives, including decimals and fractions!! Multiplication Rules 1. (6)(-4) -24 2. -4c (5d) -20cd 3. (-1)(-12) 4. 12 (-3) 5. - 2 -12 -11 -36 6. (7)(-7) 22 -49 Same signs “+” 7. 8 (-3)(2) 8. -2 -6 -4 Diff. signs “–” -48 -48 Even “-” signs “+” Odd “-” signs “–” 9. (-3b)(-4b)(-2z)(-5z)(-1) -120 b2z2 Does the number of negative values(signs) affect the sign of the product? How? Multiplying Decimals What is the rule for multiplying decimals? Line up at the right and multiply as whole number, when finished, place decimal point from right for the total number of decimal digits. Try these: Don’t forget the sign!! 10. (2.4)(-3.1) 11. 4.2(1.14) 12. (-5.3)(-0.04) -7.44 4.788 0.212 13. (2.93)(0.012) 14. (-10)(3.56)(-2) 0.03516 71.2 Multiplying Fractions What is the rule for multiplying fractions? or vertically cancel the Diagonally greatest common factor between any numerator and denominate and then multiply the numerators and denominates. What should you do to make whole numbers look like fractions? Just put a dummy denominate “1”. Try these: Don’t forget the sign!! 6 33 7 6 3 15. 2 3 16. 11 18 17. 12 7 4 3 5 2 1 3 5 8 3 1 5 18. 5 2 19. 6n 3c 5 2 24c 3 15 n 4 Properties of Multiplication Properties of Multiplication Definition Example Commutative a·b=b·a 4 · (-2) = (-2) · 4 Associative (a · b) · c = a ·(b · c) [(-4) · 3] · (-2) = (-4) · [ 3 · (-2)] Identity a·1=a (-2) · 1 = -2 Ex. (2) (–x) = –2x 3 (–n)(–n) = 3n2 –(y)4 = –(y·y·y·y) = –y4 Summary Multiplying Fractions Decimals Whole Numbers I Straight Across Ignore Decimals Count Numbers Behind Decimals To Figure Out Negative Positive Positive Negative Where Decimal X X X X Goes Negative Positve Negative Positive POSITIVE! NEGATIVE! 2.6 The Distributive Property Objective: To use the distributive property and combine like terms The distributive property: The mailman property a (b + c) = ab + ac -5 (x + 2) = (b + c) a = ab + ac (x + 4) 8 = a (b – c) = ab - ac -4 (x – 1) = (b – c) a = ab - ac (x – 5) 9 = You try these! 1. 2(x + 5)= 2 x 10 5. (x - 4)x= x 2 4 x 2. (15+6x) 1 x = 5x 2 x 2 1 6. 2 y(2 - 6y)= y 3y2 3 3. -3(x + 4)= 3x 12 7. (y + 5)(-4)= 4 y 20 4. -(6 - 3x)= 3x 6 8. 2 x 3x 9 2 x 2 6 x 3 Terms in an expression that have the same variable What are like terms? raised to the same power. 2 2 Give some examples: x y, 4 x 2 y, 0.035 yx 2 , 3 Simplified Expression: expression with no grouping symbols and all like terms combined. What does this mean you need to do? combine all like terms and possibly apply the distributive properties. What are the like terms? Combine them. 1. 8x + 3x = 11x 2. 4n2 + 2m2 – m2 = 4n 2 m 2 3. 2n2 – 5n + 7n = 2n 2 2n 4. 4a – a + 5y = 3a 5 y 5. -5x – 7x + 3x2 = 12 x 3x 2 6. x2 – xy + y2 = x 2 xy y 2 What do you do if there are parenthesis? 1. 2a(a – 3) + 4(a – 2) 2. 3x – 2(5x + 1) = 2a 2 2a 8 = 7 x 2 3. 10 – (x – 3) + 4(x2 – 3x) 4. 2w(w – 5) – (w + 6) – 4w2 = 4 x 2 13x 13 = 2w2 11w 6 You try these!! 5. –3(3m – 5) + 2m(m – 1) 6. x(4x + 2) + 5x(x – 2) = 2m2 11m 15 = 9 x2 8x Find the perimeter of a triangle (add all sides) 3x + 4 Perimeter = 3x + 4 + (2x – 3) + (x + 2) x+2 = 6x + 3 2x - 3 Summary 1. Like Terms are the terms in an expression that have the same variable raised to the same power. 2. Distributive property is the “Mailman” property. Every number in the parenthesis must multiply the “Mailman” number. 3. After using the distributive property, be aware of the operations between the terms. 2.7 Division of Real Numbers Objective: To divide real numbers Division uses the same rules to determine signs as for multiplication. Need to know: Reciprocal: If the product of a nonzero number a and another number b is 1, then a and b are ~ to each other. Note – the reciprocal of a number keeps the same sign. Give the reciprocal of the following: 2 5 1 1) 2 2) 3) 1 4) 2 5) 0 6) 3 1 3 3 2 2 2 2 2 1 5 Undef. 7 To divide real numbers Change the divisor to its reciprocal Re-write each division problem as a multiplication problem: Should the answer be positive or negative? How do you know? 2 7) 3 6 1 2 8) 6 5 -6/5 9) 1 1 3/10 3 5 2 2 4 10) 7 -1/(14x) 11) 8/a 12) 3 -1/18 4 x a 2 12 32 x 8 9 27k 13) 39 ( 4 ) 1 14) 15) 3 3 4 9 8x – 2 – 3 + 9x YOU TRY THESE!! Simplify each expression: d 3 2d 1) 6t 12t 1 2 2) 56 2 - 20 4 5 3) 4 8 3 8 x 2w 49w 36 4) - x2 5) 7 - 6) - 54 8 x 7 2 2 3 Simplify for the given value: 2x 5 2m 5n 1) and x 10 2) and m 1 and n 4 2 9 m 20 - 5 15 5 1 - 20 19 1 = - = -38 1 9 9 3 2 2 20 4 x x 2 16 3) and x 3 4) and x 4 x3 x 20 + 12 32 16 - 16 0 = = undefined = =0 -3 + 3 0 -4 -4 Summary 1. What is the reciprocal of a nonzero number? 2. What conversions do we do when dividing a nonzero number? 3. Can you determine the sign of the answer?

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