1. Warm-Up:
Describe the similarities and differences between
equations and inequalities.
Name:
Date:
Period:
Topic: Solving Absolute Value Equations & Inequalities
Essential Question: What is the process needed to solve
absolute value equations and inequalities?

2. Home-Learning #2 Review

3. Quiz #7:

4. Recall :
Absolute value | x | : is the distance
between x and 0. If | x | = 8, then
– 8 and 8 is a solution of the
equation ; or | x |  8, then any
number between 8 and 8 is a
solution of the inequality.

5. Absolute Value (of x)
• Symbol lxl
• The distance x is from 0 on the number line.
• Always positive
• Ex: l-3l=3
-4 -3 -2 -1 0 1 2
You can solve some absolute-value
equations using mental math. For
instance, you learned that the equation
| x | 3 has two solutions: 3 and 3.
To solve absolute-value equations, you
can use the fact that the expression inside
the absolute value symbols can be either
positive or negative.
Recall:

6. Solving an Absolute-Value Equation:
Solve | x  2 |  5 Solve | 2x  7 |  5  4

7. Solving an Absolute-Value Equation
| 7  2 |  | 5 |  5 | 3  2 |  | 5 |  5
Solve | x  2 |  5
The expression x  2 can be equal to 5 or 5.
x  2 IS POSITIVE
| x  2 |  5
x  2  5
x  7 x  3
x  2 IS NEGATIVE
| x  2 |  5
x  2  5
The equation has two solutions: 7 and –3.
CHECK
Answer ::

8. Solve | 2x  7 |  5  4
2x  7 IS POSITIVE
| 2x  7 |  5  4
| 2x  7 |  9
2x  7  +9
2x  16
2x  7 IS NEGATIVE
| 2x  7 |  5  4
| 2x  7 |  9
2x  7  9
2x  2
x  1
Isolate the absolute value expression on one side of the equation.
Isolate the absolute value expression on one side of the equation.
SOLUTION
2x  7 IS POSITIVE
2x  7  +9
2x  7 IS NEGATIVE
2x  7  9
2x  7 IS POSITIVE
| 2x  7 |  5  4
| 2x  7 |  9
2x  7  +9
2x  16
2x  7 IS NEGATIVE
| 2x  7 |  5  4
| 2x  7 |  9
2x  7  9
2x  2
TWO SOLUTIONS
x  8
x  1
Answer ::

9. Solve the following Absolute-Value Equation:
Practice:
1) Solve 6x-3 = 15
2) Solve 2x + 7 -3 = 8

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

11. 2) Solve 2x + 7 -3 = 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.
Answer ::

12. ***Important NOTE***
3 2x + 9 +12 = 10
- 12 - 12
3 2x + 9 = - 2
3 3
2x + 9 = - 2
3
What about this absolute value equation? 3x – 6 – 5 = – 7

14. Solving an Absolute Value Inequality:
● Step 1: Rewrite the inequality as a conjunction or a
disjunction.
● If you have a you are working with a
conjunction or an ‘and’ statement.
Remember: “Less thand”
● If you have a you are working with a
disjunction or an ‘or’ statement.
Remember: “Greator”
● Step 2: In the second equation you must negate the
right hand side and reverse the direction of the
inequality sign.
● Solve as a compound inequality.
or
 
or
 

15. Ex: “and” inequality
• Becomes an “and” problem
21
9
4 

x
-3 7 8
4x – 9 ≤ 21
+ 9 + 9
4x ≤ 30
x ≤ 7.5
4 4
4x – 9 ≥ -21
+ 9 + 9
4x ≥ -12
x ≥ -3
4 4
Positive Negative

16. |2x + 1| > 7
This is an ‘or’
statement.
(Greator).
3
-4
Ex: “or” inequality
2x + 1 > 7 or 2x + 1 < - 7
– 1 - 1 – 1 - 1
2x > 6 2x < - 8
2 2 2 2
x > 3
In the 2nd
inequality, reverse
the inequality sign
and negate the
right side value.

17. Solve | x  4 | < 3 and graph the solution.
Solving Absolute Value Inequalities:
Solve | 2x  1 | 3  6 and graph the solution.

18. Solve | x  4 | < 3
x  4 IS POSITIVE x  4 IS NEGATIVE
| x  4 |  3
x  4  3
x  7
| x  4 |  3
x  4  3
x  1
Reverse
inequality symbol.

This can be written as 1  x  7.
The solution is all real numbers greater than 1 and less than 7.
Answer ::

19. Solve | 2x  1 | 3  6 and graph the solution.
| 2x  1 |  3  6
| 2x  1 |  9
2x  1  +9
x  4
2x  8
| 2x  1 | 3  6
| 2x  1 |  9
2x  1  9
2x  10
x  5
2x + 1 IS POSITIVE 2x + 1 IS NEGATIVE
 6  5  4  3  2  1 0 1 2 3 4 5 6
The solution is all real numbers greater than or equal
to 4 or less than or equal to  5. This can be written as the
compound inequality x   5 or x  4.
Reverse
inequality
symbol.
Answer ::

20. 11
3
2
3 


x
3)
Solve and graph the following
Absolute-Value Inequalities:
4) |x -5| < 3

21. Solve & graph.
• Get absolute value by itself first.
• Becomes an “or” problem
11
3
2
3 


x
8
2
3 

x
8
2
3
or
8
2
3 



 x
x
6
3
or
10
3 

 x
x
2
or
3
10


 x
x
-2 3 4
Answer ::
3)

22. 4) |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
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.
8
2
Answer ::

23. Solve and Graph
5) 4m - 5 > 7 or 4m - 5 < - 9
6) 3 < x - 2 < 7
7) |y – 3| > 1
8) |p + 2| + 4 < 10
9) |3t - 2| + 6 = 2

24. Home-Learning #3:
• Page 211 - 212 (18, 26,36, 40, 64)