1. 4-6 Solving Absolute Value Equations & Inequalities

2. Absolute Value (of x) Symbol
Represents the distance x is from zero on the number line.
Always positive
Ex:

3. So there are two answers!
Ex:
What are the possible values of x?
x = or x = -5
So there are two answers!

4. Solving an absolute value equation:
ax+b = c, where c>0 This problem will also have two answers! Set up 2 new equations, then solve. ax+b = c and ax+b = -c

5. NB: Make sure the absolute value is by itself before you split to solve
If the absolute value isn’t by itself, solve until it is.
Ex.
Don’t set up 2 new equations until the absolute value is by itself!
isn’t the same as

6. Ex: To solve set up two equations…
6x-3 = or 6x-3 = -15
Add 3 to both sides:
6x = or 6x = -12
Divide both sides by 6
x = 3 or x = -2
* Plug in answers to check your solutions!

7. Get the absolute value part by itself by adding 3 to both sides!
Ex: To solve
Get the absolute value part by itself by adding 3 to both sides!
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
Add (-7) to both sides
2x = 4 or 2x = -18
Divide both sides by 2
x = 2 or x = -9
Check the solutions.

8. Absolute Value Inequalities!
You can also solve:
Absolute Value Inequalities!

9. The method for solving Absolute Value Inequalities depends on the inequality symbol.
Think about what this means…what values for x make this statement true?
Let’s plot them on a number line.
If it’s a less than problem, change it into an “and” compound inequality.
means AND
Another way to write this is…..

10. The method for solving Absolute Value Inequalities depends on the inequality symbol.
If it’s a less than problem, change it into an “and” compound inequality.
To write it algebraically:

11. The method for solving Absolute Value Inequalities depends on the inequality symbol.
If it’s a greater than problem, change it into an “or” compound inequality.
Think about what this means…what values for x make this statement true?
Let’s plot them on a number line.
If it’s a greater than problem, change it into an “or” compound inequality.
means OR

12. The method for solving Absolute Value Inequalities depends on the inequality symbol.
If it’s a greater than problem, change it into an “or” compound inequality. OR
To write it algebraically:
Becomes an “or” problem
Change to: ax+b > c or ax+b < -c

13. Solve & graph. Less than means it becomes an “and” problem * Add 3
* Divide by 2

14. Solve & graph. Add -3 Add 2 Divide by 2
Get absolute value by itself first.
Add -3
Since it’s greater than, it becomes an “or” problem
Add 2
Divide by 2