Absolute Value Equations and Inequalities€¦ · Absolute Value De nition 1 The absolute value of any number x is the distance between x and the zero. We denote it by jxj. Example

Absolute Value Equations and Inequalities

Dr. Abdulla Eid

College of Science

TC2MA243: Topics in Algebra

Dr. Abdulla Eid (University of Bahrain) 1 / 1

Learning Objectives:

Solve equations involving absolute values.

Solve inequality involving absolute values.


Learning Objectives:

Solve equations involving absolute values.

Solve inequality involving absolute values.


Absolute Value

Definition 1

The absolute value of any number x is the distance between x and thezero.

We denote it by |x |.

Example 2

|2| = distance between 2 and 0 = 2.

| − 3| = distance between -3 and 0 = 3.



Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.


Absolute Value

Definition 1

The absolute value of any number x is the distance between x and thezero. We denote it by |x |.

Example 2







Absolute Value

Definition 1


Example 2

|2| =

distance between 2 and 0 = 2.






Absolute Value

Definition 1


Example 2

|2| = distance between 2 and 0

= 2.






Absolute Value

Definition 1


Example 2







Absolute Value

Definition 1


Example 2


| − 3| =

distance between -3 and 0 = 3.





Absolute Value

Definition 1


Example 2


| − 3| = distance between -3 and 0

= 3.





Absolute Value

Definition 1


Example 2







Absolute Value

Definition 1


Example 2



|0| =

distance between 0 and 0 = 0.




Absolute Value

Definition 1


Example 2



|0| = distance between 0 and 0

= 0.




Absolute Value

Definition 1


Example 2







Absolute Value

Definition 1


Example 2




| − 2| =

distance between -2 and 0 = 2.



Absolute Value

Definition 1


Example 2




| − 2| = distance between -2 and 0

= 2.



Absolute Value

Definition 1


Example 2







Absolute Value

Definition 1


Example 2







Properties of Absolute Values

1 |ab| = |a| · |b|.

2

∣∣ ab

∣∣ = |a||b| .

3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.



1 |ab| = |a| · |b|.

2

∣∣ ab

∣∣ = |a||b| .

3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.



1 |ab| = |a| · |b|.

2

∣∣ ab

∣∣ = |a||b| .

3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.



1 |ab| = |a| · |b|.

2

∣∣ ab

∣∣ = |a||b| .

3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.



1 |ab| = |a| · |b|.

2

∣∣ ab

∣∣ = |a||b| .

3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.



1 |ab| = |a| · |b|.

2

∣∣ ab

∣∣ = |a||b| .

3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.


Rules

For equations (or inequalities) that involve absolute value we need to getrid of the absolute value which can be done only using the following threerules:

1 Rule 1: |X | = a→ X = a or X = −a.

2 Rule 2: |X | < a→ −a < X < a.

3 Rule 3: |X | > a→ X > a or X < −a.


Rules


1 Rule 1: |X | = a→ X = a or X = −a.

2 Rule 2: |X | < a→ −a < X < a.

3 Rule 3: |X | > a→ X > a or X < −a.


Rules


1 Rule 1: |X | = a→ X = a or X = −a.

2 Rule 2: |X | < a→ −a < X < a.

3 Rule 3: |X | > a→ X > a or X < −a.


Rules


1 Rule 1: |X | = a→ X = a or X = −a.

2 Rule 2: |X | < a→ −a < X < a.

3 Rule 3: |X | > a→ X > a or X < −a.


Rules


1 Rule 1: |X | = a→ X = a or X = −a.

2 Rule 2: |X | < a→ −a < X < a.

3 Rule 3: |X | > a→ X > a or X < −a.


Example 3

Solve |x − 3| = 2

Solution: We solve the absolute value using rule 1 to get rid of theabsolute value.

|x − 3| = 2

x − 3 = 2 or x − 3 = −2x = 5 or x = 1

Solution Set = {5, 1}.


Example 3

Solve |x − 3| = 2


|x − 3| = 2

x − 3 = 2 or x − 3 = −2x = 5 or x = 1



Example 3

Solve |x − 3| = 2


|x − 3| = 2

x − 3 = 2 or x − 3 = −2

x = 5 or x = 1



Example 3

Solve |x − 3| = 2


|x − 3| = 2

x − 3 = 2 or x − 3 = −2x = 5 or x = 1



Example 3

Solve |x − 3| = 2


|x − 3| = 2

x − 3 = 2 or x − 3 = −2x = 5 or x = 1



Exercise 4

Solve |7− 3x | = 5


Example 5

Solve |x − 4| = −3

Solution: Caution: The absolute value can never be negative, so in thisexample, we have to stop and we say there are no solutions!Solution Set = {} = ∅.


Example 5

Solve |x − 4| = −3

Solution: Caution: The absolute value can never be negative, so in thisexample, we have to stop and we say there are no solutions!

Solution Set = {} = ∅.


Example 5

Solve |x − 4| = −3

Solution: Caution: The absolute value can never be negative, so in thisexample, we have to stop and we say there are no solutions!Solution Set = {} = ∅.


Exercise 6

(Old Exam Question) Solve |7x + 2| = 16.


Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6


Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6


Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6


Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6


Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6


The Solution

1- Set notation

Solution Set = {x | − 2 < x < 6}

2- Number Line notation

3- Interval notation(−2, 6)


The Solution

1- Set notationSolution Set = {x | − 2 < x < 6}




The Solution





The Solution



3- Interval notation

(−2, 6)


The Solution





Exercise 8

Solve |3− 2x | ≤ 5


Example 9

Solve |x + 5| ≤ −2

Solution: Caution: The absolute value can never be negative or less than anegative, so in this example, we have to stop and we say there are nosolutions!Solution Set = {} = ∅.


Example 9

Solve |x + 5| ≤ −2

Solution: Caution: The absolute value can never be negative or less than anegative, so in this example, we have to stop and we say there are nosolutions!

Solution Set = {} = ∅.


Example 9

Solve |x + 5| ≤ −2

Solution: Caution: The absolute value can never be negative or less than anegative, so in this example, we have to stop and we say there are nosolutions!Solution Set = {} = ∅.


Exercise 10

(Old Final Exam Question) Solve |5− 6x | ≤ 1


Exercise 11

(Old Final Exam Question) Solve |2x − 7| ≤ 9


Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

x + 5 ≥ 7 or x + 5 ≤ −7x ≥ 2 or x ≤ −12


Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

x + 5 ≥ 7 or x + 5 ≤ −7x ≥ 2 or x ≤ −12


Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

x + 5 ≥ 7 or x + 5 ≤ −7

x ≥ 2 or x ≤ −12


Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

x + 5 ≥ 7 or x + 5 ≤ −7x ≥ 2 or x ≤ −12


The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≤ −12}


3- Interval notation(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.


The Solution

1- Set notation






The Solution

1- Set notation






The Solution

1- Set notation




(−∞,−12]∪[2,∞)



The Solution

1- Set notation






The Solution

1- Set notation






Exercise 13

Solve |3x − 4| > 1


Example 14

Solve | 3x−82 | ≥ 4

Solution:

|3x − 8

2| ≥ 4

3x − 8

2≥ 4 or

3x − 8

2≤ −4

3x − 8 ≥ 8 or 3x − 8 ≤ −83x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0


Example 14

Solve | 3x−82 | ≥ 4

Solution:

|3x − 8

2| ≥ 4

3x − 8

2≥ 4 or

3x − 8

2≤ −4

3x − 8 ≥ 8 or 3x − 8 ≤ −83x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0


Example 14

Solve | 3x−82 | ≥ 4

Solution:

|3x − 8

2| ≥ 4

3x − 8

2≥ 4 or

3x − 8

2≤ −4

3x − 8 ≥ 8 or 3x − 8 ≤ −83x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0


Example 14

Solve | 3x−82 | ≥ 4

Solution:

|3x − 8

2| ≥ 4

3x − 8

2≥ 4 or

3x − 8

2≤ −4

3x − 8 ≥ 8 or 3x − 8 ≤ −8

3x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0


Example 14

Solve | 3x−82 | ≥ 4

Solution:

|3x − 8

2| ≥ 4

3x − 8

2≥ 4 or

3x − 8

2≤ −4

3x − 8 ≥ 8 or 3x − 8 ≤ −83x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0


Example 14

Solve | 3x−82 | ≥ 4

Solution:

|3x − 8

2| ≥ 4

3x − 8

2≥ 4 or

3x − 8

2≤ −4

3x − 8 ≥ 8 or 3x − 8 ≤ −83x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0


The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≤ 0}



(−∞, 0] ∪ [16

3,∞)


The Solution

1- Set notation


3or x ≤ 0}



(−∞, 0] ∪ [16

3,∞)


The Solution

1- Set notation


3or x ≤ 0}



(−∞, 0] ∪ [16

3,∞)


The Solution

1- Set notation


3or x ≤ 0}



(−∞, 0] ∪ [16

3,∞)


The Solution

1- Set notation


3or x ≤ 0}



(−∞, 0] ∪ [16

3,∞)


The Solution

1- Set notation


3or x ≤ 0}



(−∞, 0] ∪ [16

3,∞)


Exercise 15

(Old Exam Question) Solve |x + 8|+ 3 < 2


Exercise 16

(Old Exam Question) Solve |10x − 9| ≥ 11.


Documents

Absolute Value Equations and Inequalities€¦ · Absolute Value De nition 1 The absolute value of any number x is the distance between x and the zero. We denote it by jxj. Example