This document provides information about solving and graphing inequalities. It defines inequalities and the symbols used such as <, ≤, >, ≥. It explains that inequalities have solutions that satisfy the given condition, unlike equations which have specific values. The document shows how to solve different types of inequalities algebraically by adding or subtracting from both sides and how to determine the direction of the shading or dashed line when graphing the solutions on a number line. It also discusses absolute value inequalities and graphing linear inequalities in two variables.

1. Solving and
Graphing
Inequalities
CHAPTER 6 REVIEW

2. An inequality is like an equation,
but instead of an equal sign (=) it
has one of these signs:
< : less than
≤ : less than or equal to
> : greater than
≥ : greater than or equal to

3. “x < 5”
means that whatever value x
has, it must be less than 5.
Try to name ten numbers that
are less than 5!

4. Numbers less than 5 are to the left
of 5 on the number line.
0 5 10 15
-20 -15 -10 -5
-25 20 25
• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.
• There are also numbers in between the integers, like
2.5, 1/2, -7.9, etc.
• The number 5 would not be a correct answer,
though, because 5 is not less than 5.

5. “x ≥ -2”
means that whatever value x
has, it must be greater than or
equal to -2.
Try to name ten numbers that
are greater than or equal to -
2!

6. Numbers greater than -2 are to the
right of 5 on the number line.
0 5 10 15
-20 -15 -10 -5
-25 20 25
• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.
• There are also numbers in between the integers,
like -1/2, 0.2, 3.1, 5.5, etc.
• The number -2 would also be a correct answer,
because of the phrase, “or equal to”.
-2

7. Where is -1.5 on the number line?
Is it greater or less than -2?
0 5 10 15
-20 -15 -10 -5
-25 20 25
• -1.5 is between -1 and -2.
• -1 is to the right of -2.
• So -1.5 is also to the right of -2.
-2

8. Inequalities and their Graphs
5

x
5

x
7
6
3 5
4
2 8

9. Inequalities and their Graphs
Objective: To write and graph simple
inequalities with one variable

10. Inequalities and their Graphs
7
6
3 5
4
2 8
What is a good definition for Inequality?
An inequality is a statement that
two expressions are not equal

11. Inequalities and their Graphs
Terms you see and need to know to graph inequalities correctly
Notice
open
circles
< less than
> greater than

12. Inequalities and their Graphs
Terms you see and need to know to graph inequalities correctly
Notice colored in circles
≤ less than or equal to
≥ greater than or equal to

13. Inequalities and their Graphs
Let’s work a few together
3

x
3
Notice: when variable is on
left side, sign shows
direction of solution

14. Inequalities and their Graphs
Let’s work a few together
7
Notice: when variable is on
left side, sign shows
direction of solution
7

x

15. Inequalities and their Graphs
Let’s work a few together
-2
Notice: when variable is on
left side, sign shows
direction of solution
2


p
Color in
circle

16. Inequalities and their Graphs
Let’s work a few together
8
Notice: when variable is on
left side, sign shows
direction of solution
Color in circle
8

x

17. Solve an Inequality
w + 5 < 8
w + 5 + (-5) < 8 + (-5)
w < 3
All numbers less
than 3 are
solutions to this
problem!

18. More Examples
8 + r ≥ -2
8 + r + (-8) ≥ -2 + (-8)
r ≥ -10
All numbers from -10 and up (including
-10) make this problem true!

19. More Examples
x - 2 > -2
x + (-2) + (2) > -2 + (2)
x > 0
All numbers greater than 0 make this
problem true!

20. More Examples
4 + y ≤ 1
4 + y + (-4) ≤ 1 + (-4)
y ≤ -3
All numbers from -3 down (including -3)
make this problem true!

21. There is one special case.
● Sometimes you may have to reverse the
direction of the inequality sign!!
● That only happens when you
multiply or divide both sides of the
inequality by a negative number.

22. Solving by multiplication of a
negative #
Multiply each side by the same negative number
and REVERSE the inequality symbol.
4

 x Multiply by (-1).
4


x
(-1) (-1)
See the switch

23. Solving by dividing by a negative
#
Divide each side by the same negative
number and reverse the inequality symbol.
6
2 
 x
3

x
-2 -2

24. Example:
Solve: -3y + 5 >23
-5 -5
-3y > 18
-3 -3
y < -6
●Subtract 5 from each side.
●Divide each side by negative 3.
●Reverse the inequality sign.
●Graph the solution.
0
-6

25. Try these:
1.) Solve 2x + 3 > x + 5 2.)Solve - c – 11 >23
3.) Solve 3(r - 2) < 2r + 4
-x -x
x + 3 > 5
-3 -3
x > 2
+ 11 + 11
-c > 34
-1 -1
c < -34
3r – 6 < 2r + 4
-2r -2r
r – 6 < 4
+6 +6 r < 10

26. You did remember to reverse
the signs . . .
5
7
4
15 



 x
7

7

12
4
8 


 x
4

7

4
 4

2 x 3

 
Good job!

27. Example: 8
4
6
2 

 x
x
- 4x - 4x
8
6
2 

 x
+ 6 +6
14
2 
 x
-2 -2
Ring the alarm!
We divided by a
negative!
7


x
We turned the sign!

28. Solving and Graphing Inequalities
Very Basics of Graphing Inequalities (on a number
line)
https://www.youtube.com/watch?v=nif2PKA9bXA
Graphing an inequality with the variable on the
right side and negative
https://www.youtube.com/watch?v=Em_Taf3_aRo

29. Remember Absolute Value

30. Ex: Solve 6x-3 = 15
6x-3 = 15 or 6x-3 = -15
6x = 18 or 6x = -12
x = 3 or x = -2
* Plug in answers to check your solutions!

31. Ex: Solve 2x + 7 -3 = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.

32. Ex: Solve & graph.
• Becomes an “and” problem
21
9
4 

x
2
15
3 

 x -3 7 8

33. Solve & graph.
• Get absolute value by itself first.
• Becomes an “or” problem
11
3
2
3 


x
8
2
3 

x
8
2
3
or
8
2
3 



 x
x
6
3
or
10
3 

 x
x
2
or
3
10


 x
x
-2 3 4

34. Example 1:
● |2x + 1| > 7
● 2x + 1 > 7 or 2x + 1 >7
● 2x + 1 >7 or 2x + 1 <-7
● x > 3 or x < -4
This is an ‘or’ statement.
(Greator). Rewrite.
In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
Solve each inequality.
Graph the solution.
3
-4

35. Example 2:
● |x -5|< 3
● x -5< 3 and x -5< 3
● x -5< 3 and x -5> -3
● x < 8 and x > 2
● 2 < x < 8
This is an ‘and’ statement.
(Less thand).
Rewrite.
In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
Solve each inequality.
Graph the solution.
8
2

36. Absolute Value Inequalities
Case 1 Example: 3 5
x  
3 5
2
x
x
  
 
and
8
x
5
3
x



8
x
2 



37. Absolute Value Inequalities
Case 2 Example: 2 1 9
x  
2 1 9
2 10
5
x
x
x
  
 
 
5
x  
2 1 9
2 8
4
x
x
x
 


4
x 
OR
or

38. Absolute Value
• Answer is always positive
• Therefore the following examples
cannot happen. . .
Solutions: No solution
9
5
3x 



39. Graphing Linear Inequalities
in Two Variables
•SWBAT graph a linear
inequality in two variables
•SWBAT Model a real life
situation with a linear
inequality.

40. Some Helpful Hints
•If the sign is > or < the line is
dashed
•If the sign is  or  the line will be
solid
When dealing with just x and y.
•If the sign > or  the shading
either goes up or to the right
•If the sign is < or  the shading
either goes down or to the left

41. When dealing with slanted lines
•If it is > or  then you shade above
•If it is < or  then you shade below
the line

42. Graphing an Inequality in Two Variables
Graph x < 2
Step 1: Start by graphing
the line x = 2
Now what points
would give you less
than 2?
Since it has to be x < 2
we shade everything to
the left of the line.

43. Graphing a Linear Inequality
Sketch a graph of y  3

44. Using What We Know
Sketch a graph of x + y < 3
Step 1: Put into
slope intercept form
y <-x + 3
Step 2: Graph the
line y = -x + 3