1. How can you solve a linear inequality?
<
>
<
<
Solving Inequalities
<
Lesson Essential Questions:
What is an inequality?
How can you solve a linear inequality?
<
<
>

2. Activator (5 minutes)

3. Turn & Talk
What’s the difference between an equation and an inequality?

4. The Difference is…
Equation: x + 7 = 13 Solution: x = 6 Inequality: x < 7 Solution: the value of x must be less than 7 List at least 7 numbers that are less than 7

5. A way to represent a range of numbers for a solution set
What is an inequality?
A way to represent a range of numbers for a solution set
There are several different symbols that are used to represent inequalities

6. < : less than ≤ : less than or equal to > : greater than
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

7. Examples of Inequality Statements
5  7
5 is less than 7
means
9  2
9 is greater than 2
means
-3  4
-3 is less than or
equal to 4
means
8  8
8 is greater than or
equal to 8
means
5  7
means
5 does not equal 7

8. What are the solutions?
For each problem below, state the solution or possible solutions.
x + 1 = x + 1 > 6
y – 2 = y – 2 < 3
x < y > 7
7. x  y  2

9. Inequality Symbols
Sort the words to the right into the appropriate inequality symbol column
less than
greater than
exceeds
less than or equal to
at most
greater than or equal to
no less than
no more than
at least
fewer than
more than
<
>

10. Inequalities < > £ ³ less than fewer than greater than more than
exceeds
less than or equal to
no more than
at most
greater than or equal to
no less than
at least
Inequality symbols

11. Translation into inequality is more than, is greater than
Important Words
Sample Sentence
Symbol
Translation into inequality
is at least
Bill is at least 21 years old.
is at most
At most 5 students dropped the course
cannot exceed
earnings cannot exceed $1200.
exceed
The speed exceeds 15 mph
is less than
Spot's weight is less than 50 lb.
is more than, is greater than
Boston is more than 200 miles away.
is, is equal to
The ball is 12 lbs.

12. is more than, is greater than
Important Words
Sample Sentence
Inequality
Translation
is at least
Bill is at least 21 years old. 
b > 21
is at most
At most 5 students dropped the course
<
n < 5
cannot exceed
earnings cannot exceed $1200.
r < 1200
exceeds
The speed exceeds 15 mph
>
s > 15
is less than
Spot's weight is less than 50 lb.
w < 50
is more than, is greater than
Boston is more than 200 miles away.
d > 200
is, is equal to
The ball is 12 lbs. =
b = 12

13. Your turn (2 mins): A number is less than 3.
A number is greater than or equal to 10.
A number has to be at least 5.
John cannot spend more than $40.
The auditorium seats a maximum of people.

14. Your turn: A number is less than 3.
A number is greater than or equal to 10.
A number has to be at least 5.
John cannot spend more than $40.
The auditorium seats a maximum of people.

15. How do you graph inequalities on a number line?

16. Graphing Inequalities
When we graph an inequality on a number line we use open and closed circles to represent the number.
Symbol
Meaning
Graph
>
Is greater than
<
Is less than 
Is greater than or equal to 
Is less than or equal to

17. How to graph the solutions
> Graph any number greater than. . .
open circle, line to the right
< Graph any number less than. . .
open circle, line to the left
Graph any number greater than or equal to. . .
closed circle, line to the right
Graph any number less than or equal to. . .
closed circle, line to the left

18. x < 5 means that whatever value x has, it must be less than 5.
Try to name five numbers that are less than 5!
Be Creative!

19. Numbers less than 5 are to the left of 5 on the number line.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, π, etc.
The number 5 would not be a correct answer, though, because 5 is not less than 5.

20. X < 5 Notice that an Open circle (O) was used to graph
X < 5 because the solution DOES NOT include 5.

21. Try to name five numbers that are greater than or equal to -2
x ≥ -2
means that whatever value x has, it must be greater than or equal to -2.
Try to name five numbers that are greater than or equal to -2

22. Numbers Greater than -2 are to the Right of -2 on the number line.
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”.

23. X ≥ - 2
Notice that a Closed circle ( ) was used to graph X ≥ - 2 because the solution DOES include – 2 because of the Greater than “or equal to”.

24. Graphing Inequalities
Example # 1:
p > -2
Step 1: Need a number line to graph on.
Step 2: Decide whether we can include - 2 or not. (Open or Closed Circle?)
Step 3: Draw arrow to indicate where are all the values greater than -2. (Arrow to Left or Right?)

25. Graphing Inequalities
Example # 2:
r  -5
Step 1: Need a number line to graph on.
Step 2: Decide whether we can include - 5 or not. (Open or Closed Circle?)
Step 3: Draw arrow to indicate where are all the values less than or equal to -5. (Arrow to Left or Right?)

26. Graphing Inequalities
Example # 3
n < 8
Step 1: Need a number line to graph on.
Step 2: Decide whether we can include 8 or not. (Open or Closed Circle?)
Step 3: Draw arrow to indicate where are all the values less than 8. (Arrow to Left or Right?)

27. Graphing Inequalities
Example # 4
t  -1
Step 1: Need a number line to graph on.
Step 2: Decide whether we can include -1 or not. (Open or Closed Circle?)
Step 3: Draw arrow to indicate where are all the values greater than or equal to -1. (Arrow to Left or Right?)

28. YOUR TURN X ≥ 3 Write the inequality to represent the number line

29. Quick Check What is the algebraic inequality for this graph? X  2.5

30. TEST PREP

31. TEST PREP

32. Check Your Understanding
a. Write the inequality statement down.
b. Graph each inequality statement.
x < x  -5
y  y  0
x > c > - 4
n > x  3.5

33. INEQUALITY QUIZ: Graph the following on a number line: x  -5
4. Explain the difference between x > 7 and x  7.
5. Write an algebraic inequality for the graph.
6. As required by law, the Smyrna Diner displays a sign
stating “Maximum Occupancy: 225 persons.”
a Write an inequality to describe the situation. 
b. According to the rule, is it legal for 225 people to be in the diner? Explain.

34. How do you solve inequalities?

35. Adding and Subtracting
Is 4 < 10 ?
What happens if I add 1 to each side?
What happens if I subtract 1 from each side?
Is < ?
5 < 11 TRUE
Is < ?
3 < 9 TRUE

36. Solving Inequality Statements
Solving inequalities is very similar to solving equations which you already know how to do 
REMEMBER THE RULES!!
What you do to one side – you do to the other!
Get the variable by itself!
“Undo” the operation!
Check your answer!

37. Solving Inequalities
There are a few special things to consider with inequalities:
We need to look carefully at the inequality sign.
We also need to graph the solution set on a number line.

38. Solve an Inequality w + 5 < 8 - 5 -5 w < 3
All numbers less than 3 are solutions to this problem!
w < 3 5 10 15
-20
-15
-10 -5
-25 20 25

39. More Examples 8 + r ≥ -2 -8 -8 r ≥ -10
All numbers greater than-10 (including -10)
r ≥ -10 5 10 15
-20
-15
-10 -5
-25 20 25

40. More Examples 2x > -2 2 2 x > -1
All numbers greater than -1 make this problem true!
x > -1 5 10 15
-20
-15
-10 -5
-25 20 25

41. All numbers less than 8 (including 8)
More Examples
2h + 8 ≤ 24
2h ≤ 16
All numbers less than 8 (including 8)
h ≤ 8 5 10 15
-20
-15
-10 -5
-25 20 25

42. YOUR TURN: 2 step Inequalities are just like 2 step Equations
Remember that 1st: add or subtract 2nd: Multiply or Divide 3rd: Graph on number line 2x – 3 > 5

43. Solving Inequality Statements
Let’s Practice Together! 2.

44. x + 3 > -4 6d > 24 x > -7 2x - 8 < 14 d > 4
Your Turn….
Solve the inequality and graph the answer.
x + 3 > -4
6d > 24
2x - 8 < 14
2c + 4 < 2
x > -7
d > 4
x < 11
c < -1

45. Which way is the sign?
Inequality Inequality Operation New Inequality Inequality Direction
9 > to both sides > 7 stays same
12 > 7 Multiply both sides by > 14 stays same
24> from both sides
______ Divide both sides by 5
______ Divide both sides by -2
______ Multiply both sides by 2
______ Multiply both sides by -3
______ Divide both sides by 6
______ Multiply both sides by -7
_______ + 5 to both sides
20> 10 stays same
4 > 2 stays same
20> 10
-2 < -1 flip sign
4 > 2
-2 < -1
-4 < -2 stays same
-4 < -2
12 > 6 flip sign
12 > 6
2 > 1 stays same
2 > 1
-14 < -7 flip sign
-14 < -7
-9 < -2 stays same
What operations cause the inequality sign to flip?

46. Multiply by a Negative
What happens if I multiply by – 1 to each side of 4 < 10?
Is 4 (-1) < 10 (-1) ?
Is - 4 < - 10 ?
NO! - 4 is GREATER THAN – 10…so FLIP THE INEQUALITY!
- 4 > - 10

47. Dividing by a Negative
What happens if I divide by – 2 to each side of 4 < 10?
Is 4 < 10 ?
NO!! - 2 is GREATER THAN – 5…so FLIP THE INEQUALITY!!
- 2 > - 5
Is - 2 < - 5?

48. Rule for multiplying or dividing by a negative.
WRITE THIS DOWN!!
When multiplying or dividing both sides by a negative number “flip” or “reverse” the direction of the inequality sign.

49. To Flip or Not to Flip that is the question!
3x > x < 20

50. Up to this point solving inequalities has been the same as solving equations….
Try this one:
-2x + 4 > 8

51. Let’s try another one….
3x – 5 ≤ 1

52. TURN AND TALK When do you need to switch the inequality symbol?
NOW, JOT IT DOWN WHAT YOU AND YOUR BUDDY TALKED ABOUT IT!

53. CHECK YOUR UNDERSTANDING
Complete the 4 problems
Check your answers with your buddy
Come up to me to show me your problems for a classwork grade

54. Solving Inequality Statements

55. JOURNAL
Suppose you are going to tell someone how to solve an inequality like -4x + 7 > 43.
What steps would you tell them? Why?
How would you check your solution to an inequality like -4x + 7 > 43?

57. Equations vs Inequalities

58. JOURNAL
How is solving a linear inequality, like x + 7 > 43 similar to solving a linear equation, like -4x + 7 = 43?
How is it different?

59. TEST PREP

60. Homework

61. MATH GAMES Rags to Riches: http://www.quia.com/rr/438640.html
Genie Game: play.com/Inequality-Game.html
Countdown: inequalities.html
Arcade: =play&gameid=87

62. Websites! How do we solve Inequalities???