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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.

Dr. Abdulla Eid (University of Bahrain) 2 / 1

Learning Objectives:

Solve equations involving absolute values.

Solve inequality involving absolute values.

Dr. Abdulla Eid (University of Bahrain) 2 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

|0| =

distance between 0 and 0 = 0.

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

|0| = distance between 0 and 0

= 0.

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

| − 2| =

distance between -2 and 0 = 2.

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

| − 2| = distance between -2 and 0

= 2.

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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.

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

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

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

Dr. Abdulla Eid (University of Bahrain) 3 / 1

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|.

Dr. Abdulla Eid (University of Bahrain) 4 / 1

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|.

Dr. Abdulla Eid (University of Bahrain) 4 / 1

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|.

Dr. Abdulla Eid (University of Bahrain) 4 / 1

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|.

Dr. Abdulla Eid (University of Bahrain) 4 / 1

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|.

Dr. Abdulla Eid (University of Bahrain) 4 / 1

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|.

Dr. Abdulla Eid (University of Bahrain) 4 / 1

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.

Dr. Abdulla Eid (University of Bahrain) 5 / 1

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.

Dr. Abdulla Eid (University of Bahrain) 5 / 1

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.

Dr. Abdulla Eid (University of Bahrain) 5 / 1

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.

Dr. Abdulla Eid (University of Bahrain) 5 / 1

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.

Dr. Abdulla Eid (University of Bahrain) 5 / 1

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}.

Dr. Abdulla Eid (University of Bahrain) 6 / 1

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}.

Dr. Abdulla Eid (University of Bahrain) 6 / 1

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 = −2

x = 5 or x = 1

Solution Set = {5, 1}.

Dr. Abdulla Eid (University of Bahrain) 6 / 1

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}.

Dr. Abdulla Eid (University of Bahrain) 6 / 1

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}.

Dr. Abdulla Eid (University of Bahrain) 6 / 1

Exercise 4

Solve |7− 3x | = 5

Dr. Abdulla Eid (University of Bahrain) 7 / 1

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 = {} = ∅.

Dr. Abdulla Eid (University of Bahrain) 8 / 1

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 = {} = ∅.

Dr. Abdulla Eid (University of Bahrain) 8 / 1

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 = {} = ∅.

Dr. Abdulla Eid (University of Bahrain) 8 / 1

Exercise 6

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

Dr. Abdulla Eid (University of Bahrain) 9 / 1

Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6

Dr. Abdulla Eid (University of Bahrain) 10 / 1

Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6

Dr. Abdulla Eid (University of Bahrain) 10 / 1

Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6

Dr. Abdulla Eid (University of Bahrain) 10 / 1

Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6

Dr. Abdulla Eid (University of Bahrain) 10 / 1

Example 7

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6

Dr. Abdulla Eid (University of Bahrain) 10 / 1

The Solution

1- Set notation

Solution Set = {x | − 2 < x < 6}

2- Number Line notation

3- Interval notation(−2, 6)

Dr. Abdulla Eid (University of Bahrain) 11 / 1

The Solution

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

2- Number Line notation

3- Interval notation(−2, 6)

Dr. Abdulla Eid (University of Bahrain) 11 / 1

The Solution

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

2- Number Line notation

3- Interval notation(−2, 6)

Dr. Abdulla Eid (University of Bahrain) 11 / 1

The Solution

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

2- Number Line notation

3- Interval notation

(−2, 6)

Dr. Abdulla Eid (University of Bahrain) 11 / 1

The Solution

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

2- Number Line notation

3- Interval notation(−2, 6)

Dr. Abdulla Eid (University of Bahrain) 11 / 1

Exercise 8

Solve |3− 2x | ≤ 5

Dr. Abdulla Eid (University of Bahrain) 12 / 1

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 = {} = ∅.

Dr. Abdulla Eid (University of Bahrain) 13 / 1

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 = {} = ∅.

Dr. Abdulla Eid (University of Bahrain) 13 / 1

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 = {} = ∅.

Dr. Abdulla Eid (University of Bahrain) 13 / 1

Exercise 10

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

Dr. Abdulla Eid (University of Bahrain) 14 / 1

Exercise 11

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

Dr. Abdulla Eid (University of Bahrain) 15 / 1

Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

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

Dr. Abdulla Eid (University of Bahrain) 16 / 1

Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

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

Dr. Abdulla Eid (University of Bahrain) 16 / 1

Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

x + 5 ≥ 7 or x + 5 ≤ −7

x ≥ 2 or x ≤ −12

Dr. Abdulla Eid (University of Bahrain) 16 / 1

Example 12

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

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

Dr. Abdulla Eid (University of Bahrain) 16 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≤ −12}

2- Number Line notation

3- Interval notation(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.

Dr. Abdulla Eid (University of Bahrain) 17 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≤ −12}

2- Number Line notation

3- Interval notation(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.

Dr. Abdulla Eid (University of Bahrain) 17 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≤ −12}

2- Number Line notation

3- Interval notation(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.

Dr. Abdulla Eid (University of Bahrain) 17 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≤ −12}

2- Number Line notation

3- Interval notation

(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.

Dr. Abdulla Eid (University of Bahrain) 17 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≤ −12}

2- Number Line notation

3- Interval notation(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.

Dr. Abdulla Eid (University of Bahrain) 17 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≤ −12}

2- Number Line notation

3- Interval notation(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.

Dr. Abdulla Eid (University of Bahrain) 17 / 1

Exercise 13

Solve |3x − 4| > 1

Dr. Abdulla Eid (University of Bahrain) 18 / 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

Dr. Abdulla Eid (University of Bahrain) 19 / 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

Dr. Abdulla Eid (University of Bahrain) 19 / 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

Dr. Abdulla Eid (University of Bahrain) 19 / 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 ≤ −8

3x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0

Dr. Abdulla Eid (University of Bahrain) 19 / 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

Dr. Abdulla Eid (University of Bahrain) 19 / 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

Dr. Abdulla Eid (University of Bahrain) 19 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≤ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [16

3,∞)

Dr. Abdulla Eid (University of Bahrain) 20 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≤ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [16

3,∞)

Dr. Abdulla Eid (University of Bahrain) 20 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≤ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [16

3,∞)

Dr. Abdulla Eid (University of Bahrain) 20 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≤ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [16

3,∞)

Dr. Abdulla Eid (University of Bahrain) 20 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≤ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [16

3,∞)

Dr. Abdulla Eid (University of Bahrain) 20 / 1

The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≤ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [16

3,∞)

Dr. Abdulla Eid (University of Bahrain) 20 / 1

Exercise 15

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

Dr. Abdulla Eid (University of Bahrain) 21 / 1

Exercise 16

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

Dr. Abdulla Eid (University of Bahrain) 22 / 1