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E QUATIONS & INEQUALITIES Properties

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W HAT ARE EQUATIONS ? Equations are mathematical sentences that state two expressions are equal. Example: 2x – 5 = 3(x + 4)

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W HAT ARE THE PROPERTIES OF EQUALITY ? In order to solve equations in, you must perform operations that maintain equality on both sides of the equation using the properties of equality. Properties of Equality Reflexive property of equality Symmetric property of equality Transitive property of equality Addition property of equality Subtraction property of equality Multiplication property of equality Division property of equality Substitution property of equality

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P ROPERTIES OF EQUALITY a = a A number is equal to itself. -5 = -5 If a = b, then b = a. If numbers are equal, they will still be equal if the order is changed. If x = 2, then 2 = x Reflexive Property of Equality Symmetric Property of Equality

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P ROPERTIES OF EQUALITY If a = b and b = c, then a = c. If numbers are equal to the same number, then they are equal to each other. If a = b, then a + c = b + c. Adding the same number to both sides of an equation does not change the equality of the equation. x = 6 x + 2 = 6 + 2 Transitive Property of Equality Addition Property of Equality

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P ROPERTIES OF EQUALITY If a = b, then a − c = b − c. Subtracting the same number from both sides of an equation does not change the equality of the equation. x = 6 x − 2 = 6 − 2 If a = b and c ≠ 0, then a c = b c. Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. x = 6 x 2 = 6 2 Subtraction Property of Equality Multiplication Property of Equality

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P ROPERTIES OF EQUALITY If a = b and c ≠ 0, then a ÷ c = b ÷ c. Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation. If x = 6 then If a = b, then b may be substituted for a in any expression containing a. If two numbers are equal, then substituting one in for another does not change the equality of the equation. Division Property of Equality Substitution Property of Equality

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P ROPERTIES OF OPERATIONS (R EVIEW ) PropertyRuleExample Commutative property of addition a + b = b + a3 + 5 = 5 + 3 Commutative property of multiplication a b = b a3 5 = 5 3 Associative property of addition (a + b) + c = a + (b + c)(3 + 5) + 6 = 3 + (5 + 6) Associative property of multiplication (a b) c = a (b c)(3 5) 6 = 3 (5 6) Distributive property of multiplication over addition a (b + c) = a b + a c3 (5 + 6) = 3 5 + 3 6

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W HAT PROPERTY ????? EquationWhat property ? 3x – 12 = 15original 3x = 27 x = 9

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W HAT PROPERTY ????? EquationWhat property ? original x = -18

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W HAT ABOUT INEQUALITIES ? PropertyHow it works If a > b and b > c, then a > c.If 10 > 6 and 6 > 2, then 10 > 2. If a > b, then b < a.If 10 > 6, then 6 < 10. If a > b, then –a < – b.If 10 > 6, then –10 < –6. If a > b, then a ± c > b ± c.If 10 > 6, then 10 ± 2 > 6 ± 2. If a > b and c > 0, then a c > b c.If 10 > 6 and 2 > 0, then 8 2 > b 2. If a > b and c < 0, then a c < b c. If 10 > 6, then 10 –1 < 6 –1. If you multiply by a negative on both sides, switch the inequality sign If a > b and c > 0, then a ÷ c > b ÷ c.If 10 > 6 and 2 > 0, then 8 ÷ 2 > 6 ÷ 2. If a > b and c < 0, then a ÷ c < b ÷ c. If 10 > 6, then 10 ÷ –1 < 6 ÷ –1. If you divide by a negative on both sides, switch the inequality sign

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Laws of Exponents/Review - Multiplication of Exponents - Power of Exponents - Division of Exponents - Exponents of Zero - Negative Exponents

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Multiplication of Exponents General Rule: Specific Example:

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Power of Exponents General Rule: Specific Example:

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Division of Exponents General Rule: Specific Example:

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Exponents of Zero General Rule: Specific Example:

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Negative Exponents General Rule: and Specific Example: and

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