1. Unit 2: Absolute Value Absolute Value Equations and Inequalities
Absolute Value Inequalities
Special Cases
Absolute Value Models for Distance and Tolerance

2. – 3 3
Distance is greater than 3.
Distance is 3.
Distance is less than 3.
By definition, the equation x = 3 can be solved by finding real numbers at a distance of three units from 0. Two numbers satisfy this equation, 3 and – 3.
So the solution set is

3. Properties of Absolute Value
For any positive number b:

4. SOLVING ABSOLUTE VALUE EQUATIONS
Example 1
Solve a.
Solution
For the given expression 5 – 3x to have absolute value 12, it must represent either 12 or –12 . This requires applying Property 1, with a = 5 – 3x and b = 12.

5. Solve a. Solution or or or SOLVING ABSOLUTE VALUE EQUATIONS Example 1
Property 1 or
Subtract 5. or
Divide by – 3.

6. SOLVING ABSOLUTE VALUE EQUATIONS
Example 1
Solve a.
Solution or
Divide by – 3.
Check the solutions by substituting them in the original absolute value equation. The solution set is

7. Solve b. Solution or or or SOLVING ABSOLUTE VALUE EQUATIONS Example 1
Property 2 or or

8. Use Property 3, replacing a with 2x + 1 and b with 7.
SOLVING ABSOLUTE VALUE INEQUALITIES
Example 2
Solve a.
Solution
Use Property 3, replacing a with 2x + 1 and b with 7.
Property 3
Subtract 1 from each part.
Divide each part by 2.

9. The final inequality gives the solution set (– 4, 3).
SOLVING ABSOLUTE VALUE INEQUALITIES
Example 2
Solve a.
Solution
Divide each part by 2.
The final inequality gives the solution set (– 4, 3).

10. Solve b. Solution or or or SOLVING ABSOLUTE VALUE INEQUALITIES
Example 2
Solve b.
Solution or
Property 4
Subtract 1 from each side. or or
Divide each part by 2.

11. Solve b. Solution or SOLVING ABSOLUTE VALUE INEQUALITIES Example 2
Divide each part by 2. or

12. Solve Solution or or or Example 3 Add 1 to each side. Property 4
SOLVING AN ABSOLUTE VALUE INEQUALITY REQUIRING A TRANSFORMATION
Example 3
Solve
Solution
Add 1 to each side. or
Property 4 or
Subtract 2.
Divide by – 7; reverse the direction of each inequality. or

13. Solve Solution or Example 3
SOLVING AN ABSOLUTE VALUE INEQUALITY REQUIRING A TRANSFORMATION
Example 3
Solve
Solution
Divide by – 7; reverse the direction of each inequality. or

14. SOLVING SPECIAL CASES OF ABSOLUTE VALUE EQUATIONS AND INEQULAITIES
Example 4
Solve a.
Solution Since the absolute value of a number is always nonnegative, the inequality is always true. The solution set includes all real numbers.

15. SOLVING SPECIAL CASES OF ABSOLUTE VALUE EQUATIONS AND INEQULAITIES
Example 4
Solve b.
Solution There is no number whose absolute value is less than – 3 (or less than any negative number). The solution set is .

16. SOLVING SPECIAL CASES OF ABSOLUTE VALUE EQUATIONS AND INEQULAITIES
Example 4
Solve c.
Solution The absolute value of a number will be 0 only if that number is 0. Therefore,
is equivalent to
which has solution set {– 3}. Check by substituting into the original equation.

17. Write each statement using an absolute value inequality.
USING ABSOLUTE INEQUALITIES TO DESCRIBE DISTANCES
Example 5
Write each statement using an absolute value inequality.
a. k is no less than 5 units from 8.
Solution
Since the distance from k to 8, written
k – 8 or 8 – k , is no less than 5, the distance is greater than or equal to 5. This can be written as
or equivalently

18. Write each statement using an absolute value inequality.
USING ABSOLUTE INEQUALITIES TO DESCRIBE DISTANCES
Example 5
Write each statement using an absolute value inequality.
b. n is within .001 unit of 6.
Solution
This statement indicates that the distance between n and 6 is less than .001, written
or equivalently

19. USING ABSOLUTE VALUE TO MODEL TOLERANCE
Example 6
Suppose y = 2x + 1 and we want y to be within .01 unit of 4. For what values of x will this be true?
Write an absolute value inequality.
Solution
Substitute 2x + 1 for y.
Property 3
Add three to each part.
Divide each part by 2.