1. 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 ?

2. 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 ?

3. 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

4. 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

5. 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

6. 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.

7. 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

8. – 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

9. 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

10. 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

11. 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

12. 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