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Pre-Assessment Identify the property of equality that justifies the missing steps in solving the equation below. Equation Steps 23 = 2x – 9 Original 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 8x = 512 ?

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

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**3(4x+5) = 3x – 12 12x + 15 = 3x – 12 9x + 15 = – 12 9x = – 27 x = – 3**

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

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**4(3x + 7) ≤ 2(4x + 20) 12x + 28 ≤ 8x + 40 4x + 28 ≤ 40 4x ≤ 12 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 + 40 4x ≤ 4x ≤ x ≤ 3 5. What is the solution to the equation 8x = 512 ? 8x = 512 8x = 83 x = 3

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**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

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**If a=b and c≠0, then a∙c = b∙c If a=b and c≠0, then a÷c=b÷c**

Properties of Equality Property In symbols In words 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 a=a A number is equal to itself. If numbers are equal, they will still be equal if the order is changed. If a = b, then b = a. If numbers are equal to the same number, then they are equal to each other. If a=b and b=c, then a = c. If a = b, then a + c = b + c. Adding the same number to both sides does not change the equation. Subtracting the same number to both sides does not change the equation. If a = b, 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 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 b may be substituted for a in any expression containing a. If two numbers are the same, you can substitute one for the other.

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**3 + 8= 8 + 3 a + b = b + a a ∙ b = b ∙ a 3 ∙ 8 = 8 ∙ 3**

Property General Rule Specific example Commutative property of addition Associative property of addition Commutative property of multiplication Associative property of multiplication Distributive property of multiplication over addition 3 + 8= 8 + 3 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 ∙ c) (3 ∙ 8) ∙ 2 =3 ∙ (8∙2) a ∙(b+ c) =a∙ b +a∙ c 3 ∙(8+ 2) =3∙ 8+3∙ 2

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**– 7x + 22 = 50 ____________________ _____________ ____________________ **

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

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**Example 3:Solve the equation and explain which**

Example 3:Solve the equation and explain which properties of equalities are used. 76 = 5x – x ____________________ _____________ ____________________ _____________ ____________________ _____________ ____________________ given 76= 7x - 15 combine like terms 91 = 7x addition property of equality 13 = x division property of equality symmetric property of equality x = 13 Example 4: Solve the equation and explain which properties of equalities are used x + 3(x + 4)=28 ____________________ _____________ ____________________ given 5x + 3x + 12 = 28 distributive property 8x + 12 = 28 combine like terms 8x = 16 subtraction property of equality x = 2 division property of equality

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**Lesson 2.1.2 Solving Linear Equations **

When solving equations, first take a look at the expressions on either side of the equal sign. 1. Choose which side of the equation you would like the ___________to appear on. 2. Add or subtract the other variable from both sides of the equation using either the _______________or _________________property of equality. 3. ______________ both expressions. 4. Continue to ________________ the equation. 5. ____________________your answer. variable subtraction addition Simplify solve Check

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**Example 5: Solve the equation 5x + 9 = 2x – 36.**

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

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**Example 7: Solve the equation 2(3x + 1) = 6x + 14**

________________________ 6x – 2 = 6x + 14 – 6x – 6x False, so No solution – 2 = 14 Example 8: Solve the literal equation for b1 mult by 2 divide by h subtract b2

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