Pre-Assessment 1.Identify the property of equality that justifies the missing steps in solving the equation below. EquationSteps 23 = 2x – 9Original Equation 32 = 2x 16 = x x = 16 2.What is the solution to the equation 3(4x + 5) = 3x – 12? 3. What is the solution to the inequality ? 4. What is the solution to the inequality 4(3x + 7) ≤ 2(4x + 20)? 5. What is the solution to the equation 8 x = 512 ?

Pre-Assessment 1.Identify the property of equality that justifies the missing steps in solving the equation below. EquationSteps 23 = 2x – 9Original Equation 32 = 2x 16 = x x = 16 2.What is the solution to the equation 3(4x + 5) = 3x – 12? 3. What is the solution to the inequality ? 4. What is the solution to the inequality 4(3x + 7) ≤ 2(4x + 20)? 5. What is the solution to the equation 8 x = 512 ? Addition Property of equality Division Property of equality Symmetric Property of equality

2.What is the solution to the equation 3(4x + 5) = 3x – 12? 3. What is the solution to the inequality ? 3(4x+5) = 3x – 12 12x + 15 = 3x – 12 9x + 15 = – 12 9x = – 27 x = – 3

4. What is the solution to the inequality 4(3x + 7) ≤ 2(4x + 20)? 4(3x + 7) ≤ 2(4x + 20) 12x + 28 ≤ 8x x + 28 ≤ 40 4x ≤ 12 x ≤ 3 5. What is the solution to the equation 8 x = 512 ? 8 x = x = 8 3 x = 3

Solving Equations Section 2.1

Lesson 1: Solving Equations and Inequalities Equations are mathematical sentences that state two expressions are equal. In order to solve equations in algebra, you must perform operations that maintain equality on both sides of the equation using the _____________________________. These properties are rules that allow you to balance, manipulate, and solve equations. Properties of Equality

PropertyIn symbolsIn words Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Addition Property of Equality Properties of Equality a = a A number is equal to itself. If a = b, then b = a. If numbers are equal, they will still be equal if the order is changed. 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 does not change the equation.

PropertyIn symbolsIn words Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Properties of Equality Subtracting the same number to both sides does not change the equation. If a = b and c ≠ 0, then a∙c = b∙c Multiplying both sides by the same number does not change equation. If a = b and c ≠ 0, then a ÷ c = b ÷ c Dividing by the same number does not change the equation. If a = b, then a – c = b – c.

PropertyIn symbolsIn words Substitution Property of equality Properties of Equality If a = b, then b may be substituted for a in any expression containing a. If two numbers are the same, you can substitute one for the other.

PropertyGeneral RuleSpecific example Commutative Property Associative Property Distributive Property a + b = b + a = (a + b) + c = a + (b + c)(3 + 8) + 2 =3 + (8 + 2) a ∙ b = b ∙ a 3 ∙ 8 = 8 ∙ 3 a (b + c) = a∙ b + a∙ c3(8 + 2) = 3 ∙ ∙ 2 (a ∙ b) ∙ c = a ∙ (b ∙ c)(3 ∙ 8) ∙ 2 = 3 ∙ (8 ∙ 2)

PropertyGeneral RuleSpecific example Commutative Property of Addition Associative Property of Addition Commutative Property of Multiplication a + b = b + a3 + 8 = (a + b) + c = a + (b + c)(3 + 8) + 2 = 3 + (8 + 2) a ∙ b = b ∙ a3 ∙ 8 = 8 ∙ 3

PropertyGeneral RuleSpecific example Associative Property of Multiplication Distributive Property of Multiplication over Addition (a ∙ b) ∙ c = a ∙ (b ∙ c)(3 ∙ 8) ∙ 2 = 3 ∙ (8 ∙ 2) a ∙ (b + c) = a∙ b + a∙ c 3(8 + 2) = 3 ∙ ∙ 2

Example 1:Solve the equation and explain which properties of equalities are used. – 7x + 22 = 50 ____________________ _____________ ____________________ Example 2: Solve the equation and explain which properties of equalities are used. ____________________ _____________ ____________________ Given -7x = 28 Subtraction Property of Equality x = - 4 Division Property of Equality Given Addition Property of Equality -x = 42 Multiplication Property of Equality x = -42 Mult/Division Property of Equality

Example 3:Solve the equation and explain which properties of equalities are used. 76 = 5x – x ____________________ _____________ ____________________ Example 4: Solve the equation and explain which properties of equalities are used. 5x + 3(x + 4) = 28 ____________________ _____________ ____________________ Given 76= 7x – 15 Combine Like Terms 91 = 7xAddition Property of Equality 13 = xDivision Property of Equality x = 13 Symmetric Property of Equality Given 5x + 3x + 12 = 28 Distributive Property 8x + 12 = 28Combine Like Terms 8x = 16Subtraction Property of Equality x = 2Division Property of Equality

Example 5: Solve the equation 5x + 9 = 2x – 36. Example 6: Solve the equation 7x + 4 = – 9x 5x + 9 = 2x – 36 – 2x – 2x ________________________ 3x + 9 = – 36 – 9 = – 9 ________________________ 3x = – 45 x = – 15 7x + 4 = – 9x – 7x ________________________ 4 = – 16x – 16 – 16 ____ x = 4 – 16 x= – 1 4

Example 7: Solve the equation 2(3x + 1) = 6x + 14 Example 8: Solve the literal equation for b 1 6x – 2 = 6x + 14 – 6x ________________________ – 2 = 14 False, so No solution mult by 2 divide by h subtract b 2