1. <
>
<
<
Solving 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. What do Inequalities mean?
A mathematical sentence that uses one of the inequality symbols to state the relationship between two quantities.

4. Graphing Inequalities
When we graph an inequality on a number line we use open and closed circles to represent the number.
<
<
Plot an open circle ≤ ≥
Plot a closed circle

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

6. 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.

7. Try to name ten numbers that are greater than or equal to
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

8. Numbers greater than -2 are to the right of -2 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”.

9. Dividing by a negative means switch the sign!!
Solving Inequalities!
Solving inequalities is the same as solving equations.
There are only 2 things you need to know…
1.) If you multiply or divide by a negative number you must switch the sign.
-7x < 21
x > -3
2.) You will graph your solutions.
Dividing by a negative means switch the sign!!

10. Special Case 1: Switching the Signs
When solving inequalities, if you multiply or divide by a negative you must switch the signs.
Switching the signs:
Less than becomes Greater than < switches to >
Greater than becomes Less than > switches to <
Less than or equal to becomes Greater than or equal to
≤ switches to ≥
Greater than or equal to becomes Less than or equal to
≥ switches to ≤

11. Division Property for Inequalities
Caution! Dividing by a negative number
Same if multiplying?
Notice: Sign CHANGED

12. YES! x > 2 -5 x < -10 (-5 ) x > 2(-5) 1 -5
Multiplication Property for Inequalities
Caution! When you multiply by a negative number…
YES!
x > 2 -5
…the sign CHANGES
x < -10
(-5 ) x > 2(-5)

13. Solving an Inequality
Follow the same rules and steps that we used to solve an equation.
Always undo addition or subtraction first, then multiplication.
Remember whatever is done to one side of the inequality must be done to the other side. The goal is to get the variable by itself.

14. Solve an Inequality w + 5 < 8 - 5 -5 w < 3
w < 3
All numbers less than 3 are solutions to this problem! 3 4 5 6 -1 1 2 -2 7 8

15. More Examples 8 + r ≥ -2 -8 -8 r -10 ≥ r
All numbers greater than or equal to -10 ≥ 5 10 15
-20
-15
-10 -5
-25 20 25

16. More Examples 2x > -2 2 2 x > -1
x > -1
All numbers greater than -1 make this problem true! 1 2 3 -4 -3 -2 -1 -5 4 5

17. All numbers less than or equal to 8
More Examples
2h + 8 ≤ 24
2h ≤ 16
h ≤ 8
All numbers less than or equal to 8 8 9 10 11 4 5 6 7 3 12 13

18. More Examples -3h + 6 ≤ 24 -6 -6 -3h ≤ 18 -3 -3 h > -6
-3h ≤ 18
h > -6
You are dividing by negative 3! Flip the inequality symbol!
All numbers greater or equal to -6 -6 -5 -4 -3
-10 -9 -8 -7
-11 -2 -1

19. Your Turn…. x + 3 > -4 x > -7 6d > 24 d > 4 2x - 8 < 14
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 > -3

20. Homework
Complete lesson 4-5 # 2-22 evens