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Absolute value equations have 2 answers.

Rule 1

x = 5

5

-5

x = -5

x

| | = 5

Ex 1

| 5 | = 5

| -5 | = 5

5 = 5

5 = 5

x = -5 and 5

-3 ≤ x < 7

x is greater than or equal to -3 and less than 7

When arrows point toward each other:

Rule 2

AND

x < -5 or

x ≥ 6

Ex 3

x is less than -5 or

greater than or equal to 6

When arrows point away from each other:

Rule 3

OR

Write two equations, one with a positive and one with a negative answer.

Rule 4

| x - 2 | = 5

| x - 2 | = 5

Ex 4

x - 2 = -5

x - 2 = 5

x = 5 + 2

x = -5 + 2

x =

x =

-3

7

| x - 2 | = 5

| -3 - 2 | = 5

| 7 - 2 | = 5

| - 5 | = 5

| 5 | = 5

5 = 5

5 = 5

x = -3, 7

Flip the symbol and change the right side to the opposite for second inequality.

Rule 5

Ex 5

| 4x - 8 | ≤ 24

4x - 8 ≤ 24

4x - 8 ≥ -24

4x ≥ -24 + 8

4x ≤ 24 + 8

4x ≤ 32

4x ≥ -16

x ≤ 8

x ≥ -4

-4 ≤ x ≤ 8

Ex 6 for second inequality.

| -2x + 12 | > 6

-2x + 12 > 6

-2x + 12 < -6

-2x < -6 - 12

-2x > 6 - 12

-2x > -6

-2x < -18

x < 3

x > 9

x < 3 or x > 9

Ex 7 for second inequality.

| -7x + 21 | < -9

Absolute value problems cannot be equal or unequal to a negative number.

Rule 6

no solution

______ _____ equations have 2 answers. for second inequality.

Review

Do: 1

| 6x - 9 | ≤ 27

Assignment:

Textbook Page 259, 43 - 59 odds.

Ex 5 for second inequality.

| 3x + 6 | = 21

3x + 6 = 21

3x + 6 = -21

3x = -21 - 6

3x = 21 - 6

3x = 15

3x = -27

x = 5

x = -9

x = -9, 5

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