1. Graphing and Solving Inequalities
A-REI.B.3 =
Stirrup1/06/08 (Revised:1/3/10 DM)

2. Solving Inequities Vocabulary
Inequality – the relation between two expressions or values that are not equal or out of balance. Solve – to isolate the variable. Solution – a set of values that makes the inequality a true statement. Inverse operations – operations that “undo” each other; addition and subtraction, multiplication and division.

3. 1. Use distributive property
get the letter by itself
Steps to solve:
variable
1. Use distributive property
2. Combine like terms
3. Use inverse operations (do/undo)
★ Remember the inequality rule that switches the direction of the inequality symbol only if you multiply or divide both sides by a negative to create a true statement.
You may not have to do all the steps to solve the equation
“You know you are done solving when there is a positive 1 in front of the variable.”
Medina

4. Solve inequalities by un-doing subtraction
To Solve: Do
Undo −5 +5
𝒙 < 𝟖 … -2 -4 -6 -8
-10 10 8 6 4 2
Stirrup1/06/08 (Revised:1/3/10 DM)

5. Solve inequalities by un-doing addition
To Solve: Do
Undo +2 −2
Note: Re-write inequality with variable on the left, then graph.
★ This is not the same as switching the direction of sign because the whole inequality statement is reverse not just the sign.
𝒙 < 𝟔 … -2 -4 -6 -8
-10 10 8 6 4 2
Stirrup1/06/08 (Revised:1/3/10 DM)

6. Determine if the point is a solution
Mathematically
Graphically
When x = 4
When x = -6
-6 is a solution because -6 makes the inequality a true statement -4 4 2 -2
The close circle include 4 as a solution because 4 makes the inequality a true statement.
Algebra 2 (DM)
1/4/2009

7. Determine if the point is a solution
Mathematically
Graphically
When x = -2
When x = -6
-6 is NOT a solution because -6 makes the inequality a false statement. -4 4 2 -2
The open circle means -2 is not included as a solution because -2 makes the inequality a false statement.
Algebra 2 (DM)
1/4/2009

8. Solving Inequalities Multiplication Property: By a positive number.
To Solve: Do
Undo ÷4 •4
-32
-16 16 32
Stirrup1/06/08 (Revised:1/3/10 DM)

9. Solving Inequalities Multiplication Property: By a negative value.
To Solve: Do
Undo
÷(−4)
•(−4)
When you multiply both sides by a negative value the inequality sign switches direction.
-32
-16 16 32
Stirrup1/06/08 (Revised:1/3/10 DM)

10. Solving Inequalities Division Property: By a positive value.
To Solve: Do
Undo •5 ÷5
When you multiply or divide both sides by a positive number the inequality sign doesn’t switch direction.
-16 -8 8 16
Stirrup1/06/08 (Revised:1/3/10 DM)

11. Solving Inequalities Division Property: By a negative number.
To Solve: Do
Undo
•(−4)
÷(−4)
★When you divide both sides by a negative value the inequality sign switches direction.
-12 -6 6 12
Stirrup1/06/08 (Revised:1/3/10 DM)

12. Solving Inequalities Solve: All real #s less than 6. Do Undo •2 +5 −5÷2
Note: Re-write inequality with variable on the left, then graph.
★ This is not the same as switching the direction of sign because the whole inequality statement is reverse not just the sign.
All real #s less than 6. -6 6
Stirrup1/06/08 (Revised:1/3/10 DM)

13. Solving Inequalities Solve: All real #s greater than -8.
★ Remember when you divide both sides by a negative value the inequality sign switches direction. Do
Undo
•(−3) +6 −6
÷(−3)
All real #s greater than -8. -8 8
Stirrup1/06/08 (Revised:1/3/10 DM)

14. Solving Equations with Justification
Original problem
Combine like terms
To Solve:
Subtraction Property of Inequality
D U
• 3
+ 16
- 16
÷ 3
Simplify
Division Property of Inequality
Check solution
Medina

15. 𝑥≤−8 ⃖ Solution Check Continue with check: 𝟖𝒙−𝟓𝒙+𝟏𝟔≤−𝟖 𝟖𝒙−𝟓𝒙+𝟏𝟔≤−𝟖
𝑥≤−8  ⃖
Any value less than or equal to -8
Check Starting Point
Check any other true value
𝟖𝒙−𝟓𝒙+𝟏𝟔≤−𝟖
𝟖𝒙−𝟓𝒙+𝟏𝟔≤−𝟖
𝟖     −𝟓    +𝟏𝟔≤−𝟖
𝟖      −𝟓    +𝟏𝟔≤−𝟖 −𝟖 −𝟖
−𝟏𝟎
−𝟏𝟎
−𝟔𝟒
+𝟒𝟎
+𝟏𝟔≤−𝟖
−𝟖𝟎
+𝟓𝟎
+𝟏𝟔≤−𝟖
−𝟐𝟒
+𝟏𝟔≤−𝟖
−𝟑𝟎
+𝟏𝟔≤−𝟖 −𝟖
≤−𝟖
−𝟏𝟒
≤−𝟖
The "_" means -8 is included as a solution because -8 makes the inequality a true statement and is the starting point is correct.
The inequality symbol is pointing in the correct direction because all values less than or equal to -8 would make true statements.
Medina

16. Solving Equations with Justification
Original problem
Distributive Property
Combine like terms
To Solve:
Subtraction Property of Inequality
D U
• (-4)
+30
-30
÷ (-4)
Simplify
Division Property of Inequality
Inequality rule to make true statement
Check solution!
Medina

17. 𝑥<13 ⃖ Solution Check Continue with check: −𝟗𝒙+𝟓(𝒙+𝟔)>−𝟐𝟐
𝑥<13 
★ Remember always check in original problem. ⃖
Any value less than 13
Check Starting Point
Check any other true value
−𝟗𝒙+𝟓(𝒙+𝟔)>−𝟐𝟐
−𝟗𝒙+𝟓(𝒙+𝟔)>−𝟐𝟐
−𝟗     +𝟓( +𝟔)>−𝟐𝟐
−𝟗     +𝟓( +𝟔)>−𝟐𝟐 𝟏𝟑 𝟏𝟑 𝟏𝟎 𝟏𝟎
−𝟏𝟏𝟕
+𝟓 ( )>−𝟐𝟐
+𝟓 ( )>−𝟐𝟐 𝟏𝟗
−𝟗𝟎 𝟏𝟔
−𝟗𝟎 >−𝟐𝟐
+𝟖𝟎
−𝟏𝟏𝟕 >−𝟐𝟐
+𝟗𝟓
−𝟐𝟐
>−𝟐𝟐
−𝟏𝟎
>−𝟐𝟐
The > means 13 is not included as a solution because 13 makes the inequality a false statement, however the starting point is correct.
The inequality symbol is pointing in the correct direction because all values less than 13 would make true statements.
Medina

18. Steps to solve with variables on both sides
get the variable by itself
1. Simplify on side of the equation using distributive property then combining like terms.
2. Simplify the other side of equation using distributive property then combining like terms.
3. Get the variable on the left side of inequality-MOVE VARIABLE FIRST!!!!
4. Use inverse operations (do/undo)
★ Remember the inequality rule that reverse the direction of the inequality symbol only if you multiply or divide both sides by a negative to create a true statement.
You may not have to do all the steps to solve the equation
“You know you are done solving when there is a positive 1 in front of the variable.”
Medina

19. Solving Equations with Justification
Original problem
Combine like terms
To Solve:
Subtraction Property of Inequality
D U
Simplify
• 3
- 6
+ 6
÷ 3
Addition Property of Inequality
Simplify
Division Property of Inequality
Check solution
Medina

20. 𝑥<5 ⃖ Solution Check Continue with check: 𝟐𝒙−𝟔+𝟒𝒙<𝟑𝒙+𝟗
𝑥<5  ⃖
Any value less than 5
Check Starting Point
Check any other true value
𝟐𝒙−𝟔+𝟒𝒙<𝟑𝒙+𝟗
𝟐𝒙−𝟔+𝟒𝒙<𝟑𝒙+𝟗
𝟐     −𝟔+𝟒    <𝟑    +𝟗
𝟐     −𝟔+𝟒    <𝟑    +𝟗 𝟓 𝟓 𝟓 𝟎 𝟎 𝟎 𝟏𝟎
−𝟔 < 𝟗
+𝟐𝟎 𝟏𝟓 𝟎
−𝟔 < 𝟗 +𝟎 𝟎 𝟒
+𝟐𝟎< 𝟐𝟒 −𝟔
+𝟎< 𝟗 𝟐𝟒
<𝟐𝟒 −𝟔
<𝟗
The inequality symbol is pointing in the correct direction because all values less than 5 would make true statements.
The < means 5 is not included as a solution because 5 makes the inequality a false statement, however the starting point is correct.
Medina

21. Solving Equations with Justification
Original problem
Distributive Property
To Solve:
Combine like terms
Subtraction Property of Inequality
D U
Simplify
• (-3)
+12
- 12
÷ (-3)
Subtraction Property of Inequality
Simplify
Division Property of Inequality
Inequality rule to make true statement
Check solution!

22. 𝑥≤6 ⃖ Solution Check Continue with check: 𝟒 𝒙+𝟑 +𝟐𝒙≥𝟗𝒙−𝟔
𝑥≤6 
★ Remember always check in original problem. ⃖
Any value less than or equal to 6
Check Starting Point
Check any other true value
𝟒 𝒙+𝟑 +𝟐𝒙≥𝟗𝒙−𝟔
𝟒 𝒙+𝟑 +𝟐𝒙≥𝟗𝒙−𝟔
𝟒   +𝟑 +𝟐    ≥𝟗    −𝟔
𝟒   +𝟑 +𝟐    ≥𝟗    −𝟔 𝟔 𝟔 𝟔 𝟎 𝟎 𝟎
𝟒 ≥ −𝟔 𝟗
+𝟏𝟐 𝟓𝟒
𝟒 ≥ −𝟔 𝟑 +𝟎 𝟎 𝟑𝟔
+𝟏𝟐≥ 𝟒𝟖 𝟏𝟐
+𝟎≥ −𝟔 𝟒𝟖
≥𝟒𝟖 𝟏𝟐
≥−𝟔
The inequality symbol is pointing in the correct direction because all values less than or equal to 6 would make true statements.
The "_" means 6 is included as a solution because 6 makes the inequality a true statement and is the starting point is correct.
Medina