# Absolute Value Equations and · PDF file 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 2 j2j= distance

• View
1

0

Embed Size (px)

### Text of Absolute Value Equations and · PDF file Absolute Value De nition 1 The absolute value of any...

• 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 the zero.

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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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 the zero. 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

∣∣ a b

∣∣ = |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

∣∣ a b

∣∣ = |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

∣∣ a b

∣∣ = |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

∣∣ a b

∣∣ = |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

∣∣ a b

∣∣ = |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

∣∣ a b

∣∣ = |a||b| . 3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.

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

Documents
Documents
Documents
Documents
Documents
Technology
Documents
Documents
Documents
Technology
Documents
Documents
Documents
Documents
Documents
Documents
Education
Documents
Documents
Documents
Documents
Documents
Documents
Education
Documents
Documents
Documents
Documents
Education
Documents