Order Properties of Real Numbers PDF

1 Week 3 1. memorize the real number system; 2. tell the subset which the given number belongs; 3. identify the properties of each set of numbers; and 4. apply the p...

1 Week 3 1. memorize the real number system; 2. tell the subset which the given number belongs; 3. identify the properties of each set of numbers; and 4. apply the properties in solving for the unknown in an equation. Concept There are laws and properties of real numbers that are useful in solving problems and equations. They provide equivalent formulas in solving equations and inequalities. The following are the laws or properties that govern real numbers: 1. Closure Properties 2. Commutative Properties 3. Associative Properties 4. Identity Properties 5. Inverse Properties 6. Distributive Properties 7. Multiplicative Property of Zero 8. Addition and Subtraction Properties of Equality 9. Multiplication and Division Property of Equality 10. Substitution Property 11. Reflexive Property 12. Symmetric Property 13. Transitive Property 14. Addition and Subtraction Properties of Inequality 15. Multiplication and Division Property of Inequality 16. Trichotomy Property Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  2 The Closure Properties The closure property of addition states that every pair of real numbers has a unique (one and only one) sum which is also a real number. a + b = R a unique number Illustration If a = 4 and b = 5 Then, 4 + 5 = 9 9 is a unique real number sum of 4 and 5 and nothing else. The closure property of multiplication states that each pair of real numbers has a unique product which is also a real number. a x b = R a unique number Illustration If a = 10 and b = - 3 Then, 10 (- 3) = - 30 - 30 is a unique real number product of 10 and - 3 and nothing else. The Commutative Properties The commutative property of addition states that adding two or more numbers will give the same sum regardless of the order of the numbers. a+b=b+a Illustration If a = 5 and b = 12 Then, 5 + 12 = 12 + 5 is true Since 5 + 12 = 17 and 12 + 5 = 17 The commutative property of multiplication states that multiplying two or more numbers will give the same sum regardless of the order of the numbers. axb=bxa Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  3 Illustration If a = - 17 and b = - 2 Then, (- 17) (- 2) = (- 2) (- 17) is true Since (- 17) (- 2) = 34 and (- 2) (- 17) = 34 The Associative Properties The associative property of addition states that the sum of two or more numbers will not change regardless of the grouping of the numbers. a + (b + c) = (a + b) + c Illustration If a = 7, b = 8 and c = 3 Then, 7 + (8 + 3) = (7 + 8) + 3 is true Since 7 + (11) = 18 and (15) + 3 = 18 The associative property of multiplication states that two or more numbers will give the same product regardless of the grouping of the numbers. a (bc) = (ab) c Illustration If a = 2, b = 3 and c = 4 Then, 2 (3 4) = (2 3) 4 is true Since 2 (12) = 24 and (6) 4 = 24 The Identity Properties The sum of any real number and zero is identical to the given real number. Zero is called the identity element of addition. a+0=a Illustration If a = - 21 Then, - 21 + 0 = - 21 The product of any real number and one is identical to the given real number. One Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  4 is called the identity element of multiplication. a (1) = a Illustration If a = 23 Then, 23 (1) = 23 The Inverse Properties The sum of any real number and its negative or opposite is equal to zero. The negative or opposite of the given real number is called its additive inverse. a + (- a) = 0 Illustration If a = 53 If a = - 2 Then, 53 + (- 53) = 0 Then, - 2 + 2 = 0 - 53 is the additive inverse of 53 2 is the additive inverse of - 2 The product of any real number and its reciprocal is equal to one. The reciprocal of the given real number is called its multiplicative inverse. a (1/a) = 1 Illustration If a = 8 If a = - 8 Then, 8 (1/8) = 1 Then, - 8 (- 1/8) = 1 1/8 is the reciprocal or - 1/8 is the reciprocal or multiplicative inverse of 8 multiplicative inverse - 8 The Distributive Properties Multiplication is distributive over addition. a (b + c) = ab + bc - left-hand distribution (b + c) a = ab + ac - right –hand distribution Illustration Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  5 If a = 9, b = 7 and c = 5 If a = 9, b = 7 and c = 5 Then, 9 (7 + 5) = 9 (7) + 9 (5) Then, (7 + 5) 9 = 9 (7) + 9 (5) Since, 9 (12) = 108 and Since, 12 (9) = 108 and 63 + 45 = 108 63 + 45 = 108 This is a left-hand distribution of This is a right-hand distribution of multiplication over addition. multiplication over addition. Multiplication is distributive over subtraction. a (b - c) = ab - bc - left-hand distribution (b - c) a = ab - ac - right–hand distribution Illustration If a = 9, b = 3 and c = 4 If a = 2, b = 3 and c = 4 Then, 2 (3 - 4) = 2 (3) - 2 (4) Then, (3 - 4) 2 = 4 (2) - 4 (3) Since, 2(- 1) = - 2 and Since, -1 (2) = - 2 and 6-8=-2 8-6=-2 This is a left-hand distribution This is a right-hand distribution of of multiplication over multiplication over subtraction. subtraction. The Multiplicative Property of Zero The product of any real number and zero is zero. a (0) = 0 Illustration If a = 64 Then, 64 (0) = 0 The Addition and Subtraction Properties of Equality The addition property of equality states that adding the same number to each side of an equation will result in an equivalent equation. If a - b = c, then a – b + b = c + b, or a = c + b Illustration If a = 11, b = 7 and c = 4 11 - 7 = 4 Then, (11 – 7) + 7 = 4 + 7 is true Since, (11 – 7) + 7 = 11 and 4 + 7 = 11 Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  6 The subtraction property of equality illustrates that subtracting both sides of the equation with the same number will not change the equality. If a + b = c, then a + b – b = c – b, or a = c – b Illustration If a = 45, b = 13 and c = 58 45 + 13 = 58 Then, (45 + 13) - 13 = 58 – 13 is true Since, (45 + 13) - 13 = 45 and 58 - 13 = 45 If a = 45, b = 13 and c = 58 45 + 13 = 58 Then, (45 + 13) - 13 = 58 – 13 is true Since, (45 + 13) - 13 = 45 and 58 - 13 = 45 The Multiplication and Division Properties of Equality The multiplication property of equality states that if the same nonzero number is multiplied to each side of an equation the result is an equivalent equation. If ab = c, and b≠0, then ab (b) = c (b), or a = cb Illustration If a = 26, b = - 2 and c = - 52 26 (- 2) = - 52 Then, [(26) (– 2)] (- 2) = - 52 (- 2) is true Since, [(26) (– 2)] (- 2) = 104 and – 52 (- 2) = 104 The division property of equality illustrates that dividing both sides with the same non-zero number will not change the equality If ab = c, and b≠0, then ab / b = c / b, or a = c/b Illustration Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  7 If a = 5, b = 12 and c = 60 5 (12) = 60 Then, [(5) (12)] / (12) = 60 / (12) is true Since, [(5) (12)] / (- 12) = 5 and – 52 (-12) = 5 The Substitution Property of Equality The substitution property implies that if a is equal to a certain number b, then a can be substituted for b in any equation or the number b can be substituted by a in any equation. If a = b, and a = c, then b = c If a = b, and b = c, then a = c Illustration If a = 4 + 3, b = 7 and c = 2 + 5 4 + 3 = 7 and 4 + 3 = 2 + 5 Then 7 = 2 + 5 The Reflexive Property of Equality The reflexive property shows that a number is equal to itself. a=a Illustration If a = 26 Then, 26 = 26 The Symmetric Property of Equality The symmetric property states that interchanging the sides of the equation will not change the relationship. If a + b = c, then c = a + b Illustration If a = 45, b = - 8 and c = - 52 45 + (- 8) = 37 Then, 37 = 45 + (- 8) Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  8 The Transitive Property of Equality The transitive property of equality states that two numbers equal to the same number are equal to each other. If a = b, and b = c, then a = c Illustration If a = 73, b = 16 + 57 and c = 75 - 2 73 = 16 + 57 and 16 + 57 = 75 - 2 Then, 73 = 75 – 2 is true Since, 75 - 2 = 73 The Addition and Subtraction Properties of Inequality The addition property of inequality states that adding the same number to both sides of the inequality will not change the inequality relation. If a > b, then a + c > b + c If a < b, then a + c < b + c Illustration If a = 10, b = 6 and c = 7 If a = 18, b = 33 and c = 7 10 > 6 18 < 33 Then, 10 + 7 > 6 + 7 is true Then, 18 + 7 < 33 + 7 is true Since, 17 > 13 Since, 25 < 40 The subtraction property of inequality states that subtracting the same number to both sides of the inequality will not change the inequality relation. If a > b, then a - c > b – c If a < b, then a - c < b - c Illustration If a = 16, b = - 25 and c = 7 If a = 9, b = 18 and c = 7 16 > - 25 9 < 18 Then, 16 - 7 > - 25 - 7 is true Then, 9 - 7 < 18 - 7 is true Since, 9 > -32 Since, 2 < 11 Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  9 The Multiplication and Division Properties of Inequality The multiplication property of inequality states that multiplying both sides of the inequality with the same positive number will not change the inequality relation. If a > b, then ac > bc If a < b, then ac < bc Illustration If a = 10, b = 6 and c = 7 If a = -15, b = -3 and c = 7 10 > 6 - 15 < - 3 Then, 10 (7) > 6 (7) is true Then, - 15 (7) < - 3 (7) is true Since, 70 > 42 Since, -399 < - 21 The multiplication property of inequality states that multiplying both sides of the inequality with the same negative number will change the direction of the inequality. If a > b, then a (-c) < b (c) If a < b, then a (-c) > b (c) where c is a negative number Illustration If a = 13, b = 12 and c = - 7 If a = - 5, b = - 2 and c = - 7 13 > 12 -5 - 2 (- 7) is true Since, - 91 > - 84 Since, 35 > 14 The division property of inequality states dividing both sides of the inequality with the same positive number will not change the inequality relation. If a > b, then (a / c) > (b / c) If a > b, then (a / c) > (b / c) Illustration If a = 21, b = 14 and c = 7 If a = - 35, b = - 28 and c = 7 21 > 14 - 35 < - 28 Then, 21 / 7 > 14 / 7 is true Then, - 35 / 7 < - 28 / 7 is true Since, 3 > 2 Since, - 5 < - 4 Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  10 The division property of inequality states that dividing both sides of the inequality with the same negative number will change the direction of the inequality. If a > b, then (a / c) < (b / c) If a < b, then (a / c) > (b / c) where c is a negative number Illustration If a = 21, b = - 14 and c = - 7 If a = 77, b = 98 and c = - 7 21 > - 14 77 < 98 Then, 21 / - 7 < - 14 / - 7 is true Then, 77 / - 7 > 98 / - 7 is true Since, - 3 < 2 Since, - 11 > - 14 The Trichotomy Property The trichotomy property states that between two numbers a and b, one and only one of the following is true: the first number is less than the other; the two numbers are equal or the first number is greater than the other. For a and b, only one will hold true: a < b; a = b; or a > b Illustration Given: Given: Given: a = 17 and b = 23 a = 19 and b = 19 a = 13 and b = 5 If 17 < 23 If 19 = 19 If 13 > 5 Then, 17 ¹ 23 Then, 19 ≯ 19 Then, 13 ¹ 5 Or 17 ≯ 23 Or 19 ≮ 19 Or 13 ≮ 5 Application Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  11 1. Given: 2x + 16 = - x - 2 2x + x + 16 = - x + x - 2 ( 2 + 1 ) ( x ) + 16 = ( - 1 + 1 ) ( x ) – 2 3x + 16 = 0x - 2 3x + 16 – 16 = - 2 - 16 3x = - 18 3x - 18 = 3 3 x=-6 2. Given: x – 5 > 2x + 25. Solve for x x – 5 > 2x + 25 x - 2x – 5 > 2x – 2x + 25 ( 1 – 2 ) x – 5 > ( 2 – 2 ) x + 25 - x – 5 > 0x + 25 - x – 5 + 5 > 25 + 5 - x > 30 - x ( - 1 ) < 30 ( - 1 ) x < - 30 Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor  12 Bernabe, Julieta G, de Leon, C. Elementary Algebra. Manila: JTW Corp. 2002 Blitzer, Robert. Algebra and Trigonometry 2nd Edition. Philippines: Pearson Education, Inc. as Prentice Hall. 2004. Dayrit Benjamin C, Yap, A. Modern College Algebra. Manila: Rex Bookstore. 2002 Hart, W.L.(1953). College Algebra. United States of America: D.C. Health and Company, Inc. Leithold, Louis. College Algebra. Mandaluyong City: Addison-Wesley Publishing Co. 1991. Talamayan, C.L.& C.P. Binarao (2002). College Algebra. 705 J.P. Rizal St., Makati City, Philippines: Grandwater Publications Properties of equality https://tinyurl.com/y52ofbup Properties of equality https://tinyurl.com/y5o74mxz Properties of equality https://tinyurl.com/y9r4npp6 Properties of equality https://tinyurl.com/yy8tz4aa Properties of equality https://tinyurl.com/y3ag8xet Properties of real numbers https://tinyurl.com/y6bwyqnd Transitive Property of Equality: Definition & Example. (2015, January 19). Retrieved from https://tinyurl.com/y6x9t27r Math 101 College and Advanced Algebra [Type here] First semester, AY 2024-2025 Learning Sheets Ma. Quimar Q. Gahit - Instructor