1. Solving Inequalities
To solve an inequality, use the same procedure as solving an equation with one exception. When multiplying or dividing by a negative number, reverse the direction of the inequality sign.
-3x < divide both sides by -3
-3x/-3 > 6/-3
x > -2

2. Solutions…. You can have a range of answers……
All real numbers less than 3
x< 3

3. Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 4 5
All real numbers greater than -2
x > -2

4. Solutions continued…. -5 -4 -3 -2 -1 0 1 2 3 4 5
All real numbers less than or equal to 2

5. Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 4 5
All real numbers greater than or equal to -3

6. Did you notice, Some of the dots were solid and some were open?
Why do you think that is?
If the symbol is > or < then dot is open because it can not be equal.
If the symbol is  or  then the dot is solid, because it can be that point too.

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
-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
We will use the same steps that we did with equations, if a number is added to the variable, we add the opposite sign to both sides:
w (-5) < 8 + (-5)
w + 0 < 3
All numbers less than 3 are solutions to this problem!
w < 3

9. THE TRAP…..
When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.

10. Solving using Multiplication
Multiply each side by the same positive number.
(2)

11. Solving Using Division
Divide each side by the same positive number. 3

12. Solving by multiplication of a negative #
Multiply each side by the same negative number and REVERSE the inequality symbol.
Multiply by (-1).
(-1)
See the switch

13. Solving by dividing by a negative #
Divide each side by the same negative number and reverse the inequality symbol. -2

14. Solving Inequalities 3b - 2(b - 5) < 2(b + 4)1

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

16. All numbers from -3 down (including -3) make this problem true!
More Examples
4 + y ≤ 1
4 + y + (-4) ≤ 1 + (-4)
y + 0 ≤ -3
y ≤ -3
All numbers from -3 down (including -3) make this problem true!

17. Solving compound inequalities is easy if . . .
. . . you remember that a compound inequality is just two inequalities put together.

18. You can solve them both at the same time:

19. Write the inequality from the graph:5 10 15
-20
-15
-10 -5
-25 20 25
3: Write variable:
1: Write boundaries:
2: Write signs:

20. Is this what you did?
Solve the inequality:

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

22. Solving Absolute Value Inequalities
Solving absolute value inequalities is a combination of solving absolute value equations and inequalities.
Rewrite the absolute value inequality.
For the first equation, all you have to do is drop the absolute value bars.
For the second equation, you have to negate the right side of the inequality and reverse the inequality sign.

23. Solve: |2x + 4| > 12 2x + 4 > 12 or 2x + 4 < -124 -8

24. Solve: 2|4 - x| < 10 |4 - x| < 5 -1 < x < 9
4 - x < and x > -5
- x < x > -9
x > and x < 9
-1 < x < 9 9 -1