1. Solving and Graphing Inequalities
CHAPTER 6 REVIEW
Solving and Graphing Inequalities

2. ≥ : greater than or equal to
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.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. Try to name ten 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 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.5 10 15
-20
-15
-10 -5
-25 20 25 -2
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”.

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

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

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

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

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

12. 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.

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

14. 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
x > 2
-c > 34
c < -34
3r – 6 < 2r + 4
-2r r
r – 6 < 4
r < 10

15. You did remember to reverse the signs . . .
Good job!
. . . didn’t you?

16. Ring the alarm! We divided by a negative!
Example:
- 4x x
Ring the alarm! We divided by a negative!
We turned the sign!

17. Remember Absolute Value

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

19. Get the abs. value part by itself first!
Ex: Solve 2x = 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.

20. Ex: Solve & graph.
Becomes an “and” problem

21. Solve & graph. Get absolute value by itself first.
Becomes an “or” problem

22. Example 1: |2x + 1| > 7 2x + 1 > 7 or 2x + 1 >7
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.
|2x + 1| > 7
2x + 1 > 7 or 2x + 1 >7
2x + 1 >7 or 2x + 1 <-7
x > 3 or x < -4 3 -4

23. Example 2: 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.
|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 8 2

24. Absolute Value Inequalities
Case 1 Example:
and

25. Absolute Value Inequalities
Case 2 Example: or OR

26. Solutions: No solution
Absolute Value
Answer is always positive
Therefore the following examples
cannot happen. . .
Solutions: No solution

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

28. 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

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

30. 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.

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

32. Step 1: Put into slope intercept form
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