1. The document discusses absolute value equations and inequalities. It defines absolute value and shows how to solve absolute value equations by isolating the absolute value and setting up two equations. It also explains how to solve absolute value inequalities by writing them as compound inequalities without absolute value symbols. Examples are provided to demonstrate solving and graphing different types of absolute value equations and inequalities.

1. 1.6 ABSOLUTE VALUE
EQUATIONS AND
INEQUALITIES
Part 1: Absolute Value Equations

2. ABSOLUTE VALUE
 The absolute value of a real number x, written
|x|, is its distance from zero on the number line
 Example:|5| =5
|-5| = 5

3. ABSOLUTE VALUES EQUATION
 An absolute value equation is an equation
that has a variable inside the absolute value sign
 Absolute value equations can have two answers
because opposites have the same absolute value.

4. SOLVING ABSOLUTE VALUE
EQUATIONS
 To solve an absolute value equation:
1. Isolate the absolute value
2. Remove the absolute value signs and set up as
shown:
x =k
x = − k or x = k
3. Check your answers!

5. SOLVE EACH EQUATION.
GRAPH THE SOLUTION.
2x −1 = 5

6. SOLVE EACH EQUATION.
GRAPH THE SOLUTION.
3x + 2 = 4

7. SOLVE EACH EQUATION.
GRAPH THE SOLUTION.
3 x + 2 −1 = 8

8. SOLVE EACH EQUATION.
GRAPH THE SOLUTION.
2 x+9 +3= 7

9. SOLVE EACH EQUATION.
GRAPH THE SOLUTION.

10. EXTRANEOUS SOLUTIONS
 An extraneous solution is a solution derived from an
original equation that is not a solution of the original
equation.
 Remember that the absolute value measures the
distance from zero on a number line. Distance can
never be negative. Therefore, we must check our
answers when working with absolute values.

11. SOLVE AND CHECK FOR
EXTRANEOUS SOLUTIONS.
x + 2 = 8x

12. SOLVE AND CHECK FOR
EXTRANEOUS SOLUTIONS.
3x + 2 = 4 x + 5

13. SOLVE AND CHECK FOR
EXTRANEOUS SOLUTIONS.

14. HOMEWORK
 Page 46
 #1 –3
 #10 – 18 all
 #19 – 23 odd
 #43 – 51 odd

15. 1.6 ABSOLUTE VALUE
EQUATIONS AND
INEQUALITIES
Part 2: Absolute Value Inequalities

16. ABSOLUTE VALUE INEQUALITIES
 An absolute value inequality is an inequality
that has a variable inside the absolute value
sign.

17. SOLVING ABSOLUTE VALUE
INEQUALITIES
 Writethe absolute value inequality as a
compound inequality without absolute
value symbols
x < a
 Compound : and
x ≤ a
x > a
 Compound : or
x ≥ a

18. Page 44

19. SOLVE AND GRAPH THE SOLUTION.
2x −1 < 5

20. SOLVE AND GRAPH THE SOLUTION.
3x − 4 ≤ 8

21. SOLVE AND CHECK FOR
EXTRANEOUS SOLUTIONS.

22. SOLVE AND GRAPH THE SOLUTION.
2x + 4 ≥ 6

23. SOLVE AND GRAPH THE SOLUTION.

24. WRITE EACH COMPOUND
INEQUALITY AS AN ABSOLUTE
VALUE INEQUALITY

25. HOMEWORK
 Page 45
 #4,5
 #25 – 36 all
 #57 – 65 odd