1. Inequalities and their Graphs

2. Inequalities and their Graphs
What is an inequality?
A statement that two expressions are not equal

3. Inequalities and their Graphs
less than
greater than
<
>
Open circles

4. < = ≤ > = ≥ Inequalities and their Graphs less than or equal to
greater than or equal to
< = ≤
> = ≥
Closed circles

5. Inequalities and their Graphs
X is greater than 3
Open circle
Arrow points right -1 1 2 3 4 5

6. Inequalities and their Graphs
X is less than -2
Open circle
Arrow points left -6 -5 -4 -3 -2 -1

7. Inequalities and their Graphs
n is greater than or equal to 8
Closed circle
Arrow points right 5 6 7 8 9 10 11

8. Inequalities and their Graphs
p is less than or equal to 2
Closed circle
Arrow points left -2 -1 1 2 3 4

9. x ≥ 1 x > -3 x < 2 x ≤ 0 Name the Graph: -3 -2 -1 1 2 3 -3 -2 -11 2 3
x > -3 -3 -2 -1 1 2 3
x < 2 -3 -2 -1 1 2 3
x ≤ 0 -3 -2 -1 1 2 3

10. - 1 -1 Solving an inequality x + 1 ≤ 3 x ≤ 2 0 + 1 ≤ 3 1 ≤ 3  -3 -2
x ≤ 2 -3 -2 -1 1 2 3
Check: Choose a point included on the graph and see if its true…..
Choose x = 0, is x + 1 ≤ 3 true?
0 + 1 ≤ 3
1 ≤ 3 

11. +4 +4 Solving an inequality x - 4 ≥ -1 x ≥ 3 4 - 4 ≥ -1 0 ≥ -1  -1 1
x ≥ 3 -1 1 2 3 4 5
Check: Choose a point included on the graph and see if its true…..
Choose x = 4, is x - 4 ≥ -1 true?
4 - 4 ≥ -1
0 ≥ -1 

12. -3 -3 Solving an inequality x + 3 > 2 x > -1 0 + 3 > 2
x > -1 -3 -2 -1 1 2 3
Check: Choose a point included on the graph and see if its true…..
Choose x = 0, is x + 3 > 2 true?
0 + 3 > 2
3 > 2 

13. +4 +4 Solving an inequality x - 4 < -3 x < 1 0 – 4 < -3
x < 1 -3 -2 -1 1 2 3
Check: Choose a point included on the graph and see if its true…..
Choose x = 0, is x - 4 < -3 true?
0 – 4 < -3
-4 < -3 

14. Solving an inequality 4x < 8 4 4 x < 2 4(0) < 8 0 < 8  -3
x < 2 -3 -2 -1 1 2 3
Check: Choose a point included on the graph and see if its true…..
Choose x = 0, is 4x < 8 true?
4(0) < 8
0 < 8 

15. Solving an inequality x 2 ∙ > 1 ∙ 2 2 x > 2 4/2 > 1 2 > 1
> 1
∙ 2 2
x > 2 -1 1 2 3 4 5
Check: Choose a point included on the graph and see if its true…..
Choose x = 4, is x/2 > 1 true?
4/2 > 1
2 > 1 

16. Warning: dividing by a negative number may be different….
Solving an inequality
Warning: dividing by a negative number may be different….
-3x ≤ 6
x ≤ -2 -3 -2 -1 1 2 3
Check: Choose a point included on the graph and see if its true…..
Choose x = -3, is -3x ≤ 6 true?
-3(-3) ≤ 6 NO
9 ≤ 6

17. Solving an inequality -3x ≤ 6 -3 -3 x ≥ -2 -3(0) ≤ 6 0 ≤ 6  -3 -2 -1
Rule: When dividing (or multiplying) by a negative number, you need to switch the sign in the opposite direction
-3x ≤ 6
x ≥ -2 -3 -2 -1 1 2 3
Check: Choose a point included on the graph and see if its true…..
Choose x = 0, is -3x ≤ 6 true?
-3(0) ≤ 6
0 ≤ 6 

18. Solving an inequality x -2 ∙ > 1 ∙ -2 -2 x < -2 -4/-2 > 1
Rule: When dividing (or multiplying) by a negative number, you need to switch the sign in the opposite direction
-2 ∙
> 1
∙ -2 -2
x < -2 -6 -5 -4 -3 -2 -1
Check: Choose a point included on the graph and see if its true…..
Choose x = -4, is x/-2 > 1 true?
-4/-2 > 1
2 > 1 