1. Warm-Up x < -4 or x > 1 0 < x < 7 -1 < x < 3
Write out the compound inequalities.
x is at most -4 or at least 1
x is less than 7 and greater than or equal to 0
Solve the inequalities.
-11 < -3x – 2 < x – 3 > 5 or 3x + 2 > -1
x < -4 or x > 1
0 < x < 7
-1 < x < 3
x < -4 or x > -1

2. Solving Absolute Value Equations and Inequalities
Objectives:
Solve absolute value equations
Solve absolute value inequalities
Vocabulary: absolute value

3. Essential Understanding
The solution to an absolute value equation either has:
two solutions
one negative and one positive
If there is no value of x that makes the absolute value equation true, it has no solution.
The solution to an absolute value inequality is a compound inequality that uses And or Or.

4. Absolute Value (|x|)
the distance a number is away from the origin on a number line
The distance away from zero is less than 3 in both the positive and negative direction.

5. – 8 and 8 is a solution of the
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.

6. Absolute value – the distance a number is away from the origin on a number line1 2 3 4 5 6 7 8 9 10 11 -3 -2 -1 -4 -5 -6 -7 -8 -9
-10
-11
Simplify these expression:
|-5|
|7|
|-4 – 7|
|2(3 – 8)| 5 7 11 10

7. Solving Absolute Value Equations and Inequalities
Solve this equation: x
= 4
x = 4 and x = -4

8. Solving Absolute Value Equations and Inequalities
Solve this equation: x
= 7
x = 7 and x = -7

9. No solution! An absolute value must always be positive!
Solving Absolute Value Equations and Inequalities
Solve this equation: x
= -2
No solution! An absolute value must always be positive!

10. Solving Absolute Value Equations and Inequalities
Solve this equation:
Solving Absolute Equations:
Isolate the absolute value.
Break it into 2 problems – one positive and the other negative.
Solve both equations.
Check both answers.
|3x – 3| = 6
x = 3 and x = -1

11. Solving Absolute Value Equations and Inequalities
Solve this equation:
Solving Absolute Equations:
Isolate the absolute value.
Break it into 2 problems – one positive and the other negative.
Solve both equations.
Check both answers.
|x – 2| + 4 = 8
x = 6 and x = -2

12. No solution; an absolute value can never give a negative answer!
Example 1
Solving an Absolute Value Equation
Solve these equations:
a.)|6x – 3|= b.)|2x – 8|+ 2 = -22
No solution; an absolute value can never give a negative answer!
x = 3 and x = -2
Solving Absolute Equations:
Isolate the absolute value.
Break it into 2 problems – one positive and the other negative.
Solve both equations.
Check both answers.

13. The setup does not need absolute value symbols!
Solving an Absolute Value Inequality
Absolute inequality rules:
|x|< c, then –c < x < c “and”
|x|> c, then x < -c or x > c “or”
REMEMBER:
<, <: less thAND
>, >: greatOR
Solving Absolute Inequalities:
Isolate the absolute value.
Set up the problem appropriately: “and” or “or.”
Solve the inequalities.
Check both answers.
The setup does not need absolute value symbols!

14. Watch Solve and graph the inequality: |x|< 5 -5 < x < 5
Solving an Absolute Value Inequality
Solve and graph the inequality:
|x|< 5
Solving Absolute Inequalities:
Isolate the absolute value.
Set up the problem appropriately: “and” or “or.”
Solve the inequalities.
Check both answers.
-5 < x < 5
Absolute inequality rules:
|x|< c, then –c < x < c “and”
|x|> c, then x < -c or x > c “or”
<, <: Less thAND
>, >: GreatOR

15. Example 2 Solve and graph the inequalities: -4 < x < 7
Solving an Absolute Value Inequality
Solve and graph the inequalities:
a.) |4x – 6| < 22
b.)|3x – 9| > 18
Solving Absolute Inequalities:
Isolate the absolute value.
Set up the problem appropriately: “and” or “or.”
Solve the inequalities.
Check both answers.
-4 < x < 7
x < -3 or x > 9
Absolute inequality rules:
|x|< c, then –c < x < c “and”
|x|> c, then x < -c or x > c “or”
<, <: Less thAND
>, >: GreatOR

16. Example 3 Solve and graph the inequality: |10 – 4x|≤ 2 3 > x > 2
Solving an Absolute Value Inequality
Solve and graph the inequality:
|10 – 4x|≤ 2
Solving Absolute Inequalities:
Isolate the absolute value.
Set up the problem appropriately: “and” or “or.”
Solve the inequalities.
Check both answers.
3 > x > 2
2 < x < 3
Absolute inequality rules:
|x|< c, then –c < x < c “and”
|x|> c, then x < -c or x > c “or”
<, <: Less thAND
>, >: GreatOR

17. Example 4 Solve and graph the inequality: |x + 2| – 5 ≥ 8
Solving an Absolute Value Inequality
Solve and graph the inequality:
|x + 2| – 5 ≥ 8
Solving Absolute Inequalities:
Isolate the absolute value.
Set up the problem appropriately: “and” or “or.”
Solve the inequalities.
Check both answers.
x < -15 or x > 11
Absolute inequality rules:
|x|< c, then –c < x < c “and”
|x|> c, then x < -c or x > c “or”
<, <: Less thAND
>, >: GreatOR

18. Example 5 Solve and graph the inequality: |3x + 2| – 1 ≥ 10
Solving an Absolute Value Inequality
Solve and graph the inequality:
|3x + 2| – 1 ≥ 10
Solving Absolute Inequalities:
Isolate the absolute value.
Set up the problem appropriately: “and” or “or.”
Solve the inequalities.
Check both answers.
x < -13/3 or x > 3
Absolute inequality rules:
|x|< c, then –c < x < c “and”
|x|> c, then x < -c or x > c “or”
<, <: Less thAND
>, >: GreatOR

19. Example 6 Solve and graph the inequality: |2x + 5| – 1 < 6
Solving an Absolute Value Inequality
Solve and graph the inequality:
|2x + 5| – 1 < 6
Solving Absolute Inequalities:
Isolate the absolute value.
Set up the problem appropriately: “and” or “or.”
Solve the inequalities.
Check both answers.
-6 < x < 1
Absolute inequality rules:
|x|< c, then –c < x < c “and”
|x|> c, then x < -c or x > c “or”
<, <: Less thAND
>, >: GreatOR

20. Example 8
Apply Absolute Value Inequalities
RAINFALL The average annual rainfall in California for the last 100 years is 23 inches. However, the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California?

21. Example 9
A thermostat inside Macy’s house keeps the temperature within 3 degrees of the set temperature point. If the thermostat is set at 72 degrees Fahrenheit, what is the range of temperatures in the house?
A. {x | 70 ≤ x ≤ 74}
B. {x | 68 ≤ x ≤ 72}
C. {x | 68 ≤ x ≤ 74}
D. {x | 69 ≤ x ≤ 75}

22. Algebra 1 Connect Four Practice:
Each student will be responsible for working out every problem.
If it is your turn and you get the problem correct, put your initials in any of the bottom squares (like dropping a chip in the Connect Four board).
If it is your turn and you get the problem wrong, your partner will get a chance to “steal” your turn.
Can’t agree on an answer? Raise your hand! Do not argue/shout/get worked up.
Connect four in a row to win the game!
Show your work for each problem on a separate sheet of paper! Number the problems. This will be collected for points!