1. 12.1 Solving Two-Step Equations
How would you solve 35x = 350?
How would you solve x – 5 = 345?
How would you solve 35x – 5 = 345?
Isolate the variable – get it on one side of the equal sign by itself.
Do outside of grouping symbols first.
Do addition or subtraction inverse operations first.
Do multiplication or division inverse operations second.
Do inverse operations on both sides of the equal sign!
Check your answer.

2. 12.1 Solving Two-Step Equations
35x – 5 = 345
Add. Prop 35x – 5+ 5 =
Combine like terms 35x = 350
Div. Prop 35x ÷ 35 = 350 ÷ 35
x = 10
Check 35x – 5 = 345
35(10) -5 = 345
350 – 5 = 345
345 = 345
pp ex 1-3
p 698: 1-7

3. 12.1 Solving Two-Step Equations
Be careful about inverse operation order.
0 = 9(k – 2/3) + 33
0 = 9(k – 2/3) = 9(k – 2/3) + 33
-33 = 9(k -2/3) = 9(k – 2/3)
+2/3 +2/ = 9k – 6
-32 1/3 = 9k
÷9 ÷ = 9k
-3 16/27 = k ÷9 ÷9
-3 = k

4. 12.2 Solving Multi-Step Equations
Sometimes equations may be too complex to even solve in two steps. You need to take additional steps to simplify the problem.
Too many variables and constants:
4x + 2 – 2x + 1 = 11
Combine like terms 2x + 3 = 11
Subtraction Prop. 2x + 3 – 3 =
Combine Like terms and Division Prop. 2x ÷ 2 = 8 ÷ 2
x = 4
Check 4x + 2 – 2x + 1 = 11
4(4) + 2 – 2(4) + 1= 11
– = 11

5. 12.2 Solving Multi-Step Equations
Too complex – Distributive Property:
4(x + 2) = 12
Distributive Property 4x + 4 × 2 = 12
Simplify 4x = 12
Subtraction Prop. 4x + 8 – 8 = 12 – 8
Division Prop 4x ÷ 4 = 4 ÷ 4
x = 1
Check 4(x + 2) = 12
4(1 + 2) = 12
4(3) = 12
12 = 12
4(1) + 4(2) = 12
= 12

6. 12.3 Solving Equations with Variables on Both Sides
Some equations cannot be solved because the variable is on both sides of the equal sign.
Get the variables on one side of the equal sign, and get the constants on the other side.
4x = x
Subtraction Prop. 4x – 2x = x – 2x
Combine like terms 2x = 11
Subtraction Prop. 2x =
Division Prop. 2x ÷ 2 = 8 ÷ 2
x = 4
Check 4x = x
4(4) = (4) =
19 = 19

7. 12.3 Solving Equations with Variables on Both Sides
Get the variables on one side of the equal sign with no constants.
½x = 2½x – 7
Subtraction Prop. ½x ½x = 2½x – 7 - ½x
Simplify = 2x - 7
Addition Prop = 2x – 7 + 7
Simplify = 2x
Division Prop ÷ 2 = 2x ÷ 2
5 = x
Check ½(5) + 3 = 2½(5) – 7 =
5.5 = 5.5
Remember that fractions are division.

8. 12.4 Inequalities
Inequality: mathematical sentence made by placing an inequality sign (<, >, ≤, ≥) between two expressions that are not equal or may not be equal.
5 < 9 -9 > ≤ ≥ -3
Inequality signs: read symbol from left to right
Less than: <
Less than or equal: ≤
Greater than: >
Greater than or equal: ≥
Algebraic Inequality: an inequality with a variable.
Inequality: 3x < 9 x – 12 > -10 3x - 5 ≤ 10 -2x + 5 ≥ -3
Solution set: x < 3 x > 2 x ≤ 5 x ≤ 4
Solution set: the set of all numbers that you can substitute for the variable to make the inequality true.

9. 12.4 Inequalities
Graph of an inequality: set of points on a number line that show the solution of an inequality.
Open circle: < or >
Filled circle: ≤ or ≥
Darkened line: included values
Equivalent inequalities: inequalities that have the same solution. These can be formed by using inverse operations.
Compound inequality: the result of combining two inequalities. -1 ≤ x < 3

10. 12.5 Solving Inequalities by Adding and Subtracting
Subtraction Property of Equality: subtracting the same number from each side of an equation produces an equivalent equation.
x + 7 = -10 → x + 7 = -10
- 7 – 7
x = -17
Check: x + 7 = -10 → = -10
-10 = -10
Subtraction Property of Inequality : subtracting the same number from each side of an equation produces an equivalent inequality.
x + 7 < -10 → x + 7 < -10
x < -17 … “like” -17.1
Check: x + 7 < -10 → < -10
-10.1 < -10

11. 12.5 Solving Inequalities by Adding and Subtracting
Addition Property of Equality: adding the same number to each side of an equation produces an equivalent equation.
-9 = y -12 → -9 = y – 12
3 = x
Check: -9 = y -12 → -9 = 3 – 12
-9 = -9
Addition Property of Inequality: adding the same number to each side of an equation produces an equivalent inequality.
-9 ≥ y -12 → -9 ≥ y – 12
3 ≥ x “like” 3 or 2.9
Check: -9 ≥ y -12 → -9 ≥ 3 – ≥ 2.9 – 12
-9 ≥ ≥ -9.1

12. 12.6 Solving Inequalities by Multiplying and Dividing
Solve:
-3x = /y= -8 y/8 = 4
÷-3 ÷-3 × y × y ×8 ×8
x = = -8y y = 32
÷ ÷-8
-3 = y
-3x > /y ≥ -8 y/8 > 4
x < ≥ -8y y > 32
-3 ≤ y
Multiplication and Division Property of Inequalities:
Divide or multiply both sides by the same positive number.
Divide or multiply both sides by the same negative number, and reverse the inequality sign

13. 12.7 Solving Multi-Step Inequalities
Solve: x/5 – 15 < 10
Write inequality x/5 – 15 < 10
Add. Prop
Combine like terms x/5 < 25
Mult. Prop × × x < 125
Solve: 42 ≤ y/
Write inequality ≤ y/
Subt. Prop
Combine like terms 12 ≤ y/-9
Mult. Prop × × -9
REVERSE INEQUALITY ≥ y

14. 12.7 Solving Multi-Step Inequalities
Solve: 3x – 8 – 5x > 2
Write inequality 3x – 8 – 5x > 2
Add. Prop
Combine like terms -2x > 10
Div. Prop ÷ -2 ÷ -2
REVERSE INEQUALITY x < -5