Test your knowledge Of Properties from Chapters 1 & 2

Name the property demonstrated: If 3g = 4h, then 4h = 3g. The Symmetric Property gives the mirror image of one equation.

Name the property demonstrated: If 3g = 4h and 4h = 5j, then 5j = 3g. Of Equality Three equations that form a circular chain of steps. An entire side of an equation was replaced.

Name the property demonstrated: If 3g ≥ 4h and 4h ≥ 5j, then 3g ≥ 5j. “order” is a synonym for “inequality” Notice that the conclusion has to be in this left to right ORDER.

Name the property demonstrated: If 3g = 4h and h = 5j, then 3g = 4(5j). (Principle) Only part of the right side was replaced!

Name the property demonstrated: If 3g = 4h, then 3g + 5j = 4h + 5j. (of Equality) 5j was added to both sides of the equation.

Name the property demonstrated: If 3g + 5j = 4h + 5j, then 3g = 4h. (of Addition) or Addition Property of Equality “+ 5j” was cancelled from both sides of the equation.

Name the property demonstrated: If 3g + 5j = 4h + 7j, then 3g = 4h + 2j. (of Equality) You added the opposite of 5j to both sides, but it did not cancel out all the j’s on the right hand side.

Name the property demonstrated: If 3g > 4h, then -6g < -8h. (you must write “of order” since this is only true for inequalities) Reverse the inequality sign when you multiply or divide by a negative value.

(Subtraction Prop of Order is OK with Ms. Hardtke) Adding (or subtracting) the same constant from both sides of an inequality does not change the inequality.

Name the property demonstrated: If g and h are real numbers, then either g = h or g h. (or Trichotomy Principle in some textbooks) Simply assures us that two unique numbers cannot be placed on a real number line in more than one way in the same problem.

Multiplying by negative one produces the opposite value.

Name the property demonstrated: 3g ● 4 = 3 ● 4 ● g (of Multiplication) The “g” and “4” terms changed order. Remember: you commute home to school and then school to home.

Name the property demonstrated: 3(4h) = (3 ● 4) h (of Multiplication) The order of the terms did not change; only the parentheses moved. Remember: different terms are associating within the ( ).

Name the property demonstrated: (3g + 4h) + 5h= 3g + (4h + 5h) (of Addition) The order of the terms did not change; only the parentheses moved. Remember: different terms are associating within the ( ).

Name the property demonstrated: If 3g + 4h = 5j, then 4h + 3g = 5j (of Addition) The order of the terms changed. Remember: you commute home to school and then school to home.

Name the property demonstrated: If 3g + 4h = 5j, then 5j = 4h + 3g (of Equality) Symmetric Prop. gives the mirror image of the equation.

Name the property demonstrated: 3g ● 1 = 3g Multiplying by the identity element keeps the term “identical”

Name the property demonstrated: 3g + -3g = 0 (or Inverse Property of Addition) Inverse property because it produced the identity element of addition as the result.

Name the property demonstrated: -(g + h) = -g + -h (Note that Distributive is OK, but not the best answer and multiplication by -1 is not really shown here) The opposite sign affects each term of the sum.

Name the property demonstrated: 3g + 0 = 3g Adding the identity element keeps the term “identical”

Name the property demonstrated: -(g h) = -g ● h or -(g h) = g ● -h (Note that multiplication by -1 is not really shown here) The opposite sign affects just one factor of the property. Otherwise, two negatives would cancel each other.

(or Inverse Property of Mult.) Inverse property because it produced the identity element of multiplication as the result.

Match the Property Name to each statement. 1. ab + 0 = ab 2. 1ab = ab 3. ab = ba. 5. ab = ab A. Reflexive Prop (of Equality) B. Commutaive Property (of Mult.) C. Identity Prop. of Addition D. Identity Prop. of Mult. E. Inverse Prop of Mult. Or Prop. of Reciprocals

TRUE or FALSE? The set of integers is closed under addition. When you add two integers, the result is always an integer.

TRUE or FALSE? The set of integers is closed under division. A counter-example could be: 7 / 2 = 3.5

TRUE or FALSE? The set of natural numbers is closed under subtraction. A counter-example could be: 5 – 7 = -2

TRUE or FALSE? The set of natural numbers is closed under addition. Adding two natural (or counting) numbers always results in a natural number.

TRUE or FALSE? The set of real numbers is closed under the square root operation. Counter-example: the square root of a negative real number is not a real number.

TRUE or FALSE? The set of non-negative real numbers is closed under the square root operation. The square root of zero or of a positive real number is always a real number (either rational or irrational).

TRUE or FALSE? The set of even integers is closed under multiplication. Multiplying two even integers always results in an even integer.

TRUE or FALSE? The set of even integers is closed under addition. Adding two even integers always results in an even integer.

TRUE or FALSE? The set of even integers is closed under division. One counter-example: division by zero does not produce an even integer.

Which property is used below? If 3a(b + 7) = 0, then 3a = 0 or b + 7 = 0. Property of Opposite of a Sum Transitive Property Distributive Property Multiplicative Property of Zero Zero Product Property

Which property is used below? ½ + - ½ = 0 Addition Property of Equality Transitive Property Zero Product Property Inverse Property of Multiplication Property Of Opposites

Which property is used below? ½ + ¼ = ¼ + ½ Addition Property (of Equality) Transitive Property (of Equality) Associative Property (of Addition) Symmetric Property (of Equality) Commutative Prop (of Addition)

Addition Property (of Equality) Transitive Property (of Equality) Commutative Property (of Addition) Symmetric Property (of Equality) Associative Prop (of Addition)

Multiplication Property (of Equality) Transitive Property Inverse Property (of Multiplication) Symmetric Property (of Equality) Cancellation Prop (of Addition) Ms. H would accept Addition Prop (of Equality) as well, but it was not a choice here.

TRUE or FALSE? Subtraction of real numbers is commutative. One counter-example: 5 – 9 ≠ 9 - 5

Transitive Property of Order Transitive Property (of Equality) Commutative Property (of Addition) Symmetric Property (of Equality) Reflexive Property (of Equality)

TRUE or FALSE? For real numbers a and b, it is possible that 2a < 2b and 2a = 2b. This would contradict the Comparison or Trichotomy Principle. Note that this is different than 2a ≤ 2b which has the infinite solution set {a: a ≤ b}

Transitive Property of Order Transitive Property (of Equality) Reflexive Property (of Equality) Symmetric Property (of Equality) Commutative Property (of Addition)

Match the Property Name to each statement. 1. ¼ ¾ = ¼ + ¾ For 2 unique real numbers a and b, Exactly one of these is true: a = b or a > b or a < b. 2. For 2 unique real numbers a and b, Exactly one of these is true: a = b or a > b or a < b. 3. For 2 real numbers p and q, pq is a real number. 3. For 2 real numbers p and q, pq is a real number. 4. -(ab) = (- a) ● b or a ● (- b) 5. (7 + ¼) + ¾ = 7 + (¼ + ¾ ) A. Associative Prop (of Add.) B. Closure Property (of Mult.) C. Commutative Prop (of Add.) D. Comparison or Trichotomy Principle E. Opposite of a Product Prop.

TRUE or FALSE? If a < b and b < c, then c < a. The conclusion is out of order. This is a good reminder why Properties of Inequalities are called Properties of ORDER.

TRUE or FALSE? For any real numbers a, b and c, if a < b, then a + c < b +c. This is the Addition Prop of Order and it works whether c is positive, negative or zero.

TRUE or FALSE? For any real numbers a, b and c, if a < b, then ac < bc. This is true if c is positive, but it is false if c is zero or if c is negative.

Match the Property Name to each statement. 1. If xy = 0, then x = 0 or y = x = x = x 4. x + -x = x = 0x A. Reflexive Prop (of Equality) B. Identity Prop of Addition C. Zero Product Prop D. Multiplication Prop of Zero E. Inverse Prop of Addition or Prop of Opposites

TRUE or FALSE? By the Distributive Property 14xy – 7xz + 7x = 7x(2y – z) The right hand side should read 7x(2y – z + 1). If you factor a monomial from a trinomial, there should still be a trinomial in the parentheses.

Property of Reciprocals Multiplication Property (of Equality) Reflexive Property (of Equality) Inverse Property (of Multiplication) Multiplicative Identity Property