1. Algebra 6-5 Solving Open Sentences Involving Absolute Value
– 3 – 2 –
| | | | | | | | | |
– 5 – 4 – 3 – 2 –
| | | | | | | | | |
Harbour

2. Solving Open sentences involving absolute vale
Section 6-5
Solving Open sentences involving absolute vale

3. Solving Open Sentences Involving Absolute Value
Algebra 6-5 Solving Open Sentences Involving Absolute Value
There are three types of open sentences that can involve absolute value.
Consider the case | x | = n.
| x | = 5 means the distance between 0 and x is 5 units
If | x | = 5, then x = – 5 or x = 5.
The solution set is {– 5, 5}.
Solving Open Sentences Involving Absolute Value
Harbour

4. Algebra 6-5 Solving Open Sentences Involving Absolute Value
When solving equations that involve absolute value, there are two cases to consider:
Case 1 The value inside the absolute value symbols is positive.
Case 2 The value inside the absolute value symbols is negative.
Equations involving absolute value can be solved by graphing them on a number line or by writing them as a compound sentence and solving it.
Harbour

5. Solve an Absolute Value Equation
Method 1 Graphing
means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction.
The distance from –6 to –11 is 5 units.
The distance from –6 to –1 is 5 units.
Answer: The solution set is

6. Solve an Absolute Value Equation
Method 2 Compound Sentence
Write as or
Case 1
Case 2
Original inequality
Subtract 6 from each side.
Simplify.
Answer: The solution set is
Example 5-1a

7. Solve an Absolute Value Equation
Answer: {12, –2}

8. Write an Absolute Value Equation
Write an equation involving the absolute value for the graph.
Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1.
The distance from 1 to –4 is 5 units.
The distance from 1 to 6 is 5 units.
So, an equation is .

9. Write an Absolute Value Equation
Answer:
Check Substitute –4 and 6 into

10. Write an Absolute Value Equation
Write an equation involving the absolute value for the graph.
Answer:

11. Algebra 6-5 Solving Open Sentences Involving Absolute Value
Consider the case | x | < n.
| x | < 5 means the distance between 0 and x is LESS than 5 units
If | x | < 5, then x > – 5 and x < 5.
The solution set is {x| – 5 < x < 5}.
Harbour

12. Algebra 6-5 Solving Open Sentences Involving Absolute Value
When solving equations of the form | x | < n, find the intersection of these two cases.
Case 1 The value inside the absolute value symbols is less than the positive value of n.
Case 2 The value inside the absolute value symbols is greater than negative value of n.
Harbour

13. Solve an Absolute Value Inequality (<)
Then graph the solution set.
Write as and
Case 1
Case 2
Original inequality
Add 3 to each side.
Simplify.
Answer: The solution set is

14. Solve an Absolute Value Inequality (<)
Then graph the solution set.
Answer:

15. Algebra 6-5 Solving Open Sentences Involving Absolute Value
Consider the case | x | > n.
| x | > 5 means the distance between 0 and x is GREATER than 5 units
If | x | > 5, then x < – 5 or x > 5.
The solution set is {x| x < – 5 or x > 5}.
Harbour

16. Algebra 6-5 Solving Open Sentences Involving Absolute Value
When solving equations of the form | x | > n, find the union of these two cases.
Case 1 The value inside the absolute value symbols is greater than the positive value of n.
Case 2 The value inside the absolute value symbols is less than negative value of n.
Harbour

17. Solve an Absolute Value Inequality (>)
Then graph the solution set.
Write as or
Case 1
Case 2
Original inequality
Add 3 to each side.
Simplify.
Divide each side by 3.
Simplify.

18. Solve an Absolute Value Inequality (>)
Answer: The solution set is

19. Solve an Absolute Value Inequality (>)
Then graph the solution set.
Answer:

20. Algebra 6-5 Solving Open Sentences Involving Absolute Value
In general, there are three rules to remember when solving equations and inequalities involving absolute value:
If then or (solution set of two numbers)
If then and (intersection of inequalities)
If then or (union of inequalities)
Harbour

21. Assignment Study Guide 6-5 (In-Class)
Pages #’s 14-19, 24-35, 40, 41. (Homework)