6-6 Solving Inequalities Involving Absolute Value

Algebra 1 Glencoe McGraw-Hill Linda Stamper

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Which type of compound inequality, “AND” or “OR”, has a greater number of solutions? Why?

“Or” because it graphs as opposite rays that continue to infinity.

An “AND” type of compound inequality will graph as ....An “OR” type of compound inequality will graph as ....

An absolute-value inequality is an inequality that has one of these forms:

cbax cbax cbax cbax

When the absolute value is on the left, the Less than symbol represents the “AND” type of inequality. It graphs as a line segment and has less – fewer solutions.

When the absolute value is on the left, the Greater than symbol represents the “OR” type of inequality. It graphs as two opposite rays and has a greater number of solutions.I should

copy the above notes

in my notebook!

Identify the inequality as an “AND” type or “OR” type. Then identify the graph as a line segment or opposite rays.

4x

Inequality

Type Graph

AND line segment

92x OR opposite rays

54x3 OR opposite rays

86x3 AND line segment

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BEFORE you identify the type of compound inequality you must isolate the absolute value.

An absolute-value inequality is an inequality that has one of these forms:

cbax cbax cbax cbax

“AND” type

“OR” type

To solve an absolute-value inequality, write the two related inequalities – a positive inequality and a negative inequality.

When you write the related inequality for the negative value, reverse the inequality symbol.

Do not identify the

type of inequality until the absolute value is isolated!

Solve. Then graph the solution.

16x

Write the positive related inequality.

16x

Write the inequality.

Write the negative related inequality;

and 16x

LA46x4

Isolate the absolute value on one side of the inequality sign.

4 4

–7 –5

Solve each inequality.O O

Write as a single inequality.

5x 6 66 6

5x7

Graph.

7x and

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Do not identify the

type of inequality until the absolute value is isolated!

Solve. Then graph the solution.

16x

Write the positive related inequality.

16x

Write the inequality.

Write the negative related inequality;

LA46x4

Isolate the absolute value on one side of the inequality sign.

4 4

–7 –5Solve.O O

6 66

5x7

Graph.

I should copy the

above notes in my

notebook!

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1

Solve. Then graph the solution.

16x

Write the positive related inequality.

16x

Write the inequality.

Write the negative related inequality;

ro 16x

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46x4

Isolate the absolute value on one side of the inequality sign.

4 4

–7 –5

Solve each inequality.O O

Reposition.

5x 6 66 6

Graph.

7x

or

>/

5x 7x

Solve. Then graph the solution.

4x

Example 1 8x

Example 2

Solve. Then graph the solution.

4x

4x 4x4

–4 4

4x and

• •

Note: Less than symbol represents the “AND” type of inequality and will graph as a line segment.

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Example 1 4x

4x

–4 4

• •

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Example 1

4

Solve. Then graph the solution.

8x

8x

–8 8

8x ro

• •

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or 8x

Example 2

8x

Note: Greater than symbol represents the “OR” type of inequality and will graph as opposite rays.

Solve. Then graph the solution.

19x

Absolute value cannot be less than zero (cannot be negative)! Thus there are no values that will be less than negative one.

Solve. Then graph the solution.

19x

Absolute value will be zero or greater (it cannot be negative). Thus x can be any real number and the absolute value will be greater than negative one.

all real numbers

0•

Solve. Then graph the solution.Example 3

Example 4

Example 6

19x

153x

849x32

Example 7 52

4a3

Example 8 76x

Example 5 512x

Example 9 81x3

Example 3 Solve. Then graph the solution.

19x

19x

8 10

19x or

• •

Note: Greater than symbol represents the “OR” type of inequality and will graph as opposite rays.

or8x

10x 8x

10x or

9 99 9

GO

19x

Example 4 Solve. Then graph the solution.

153x

43x

7x

–1 7

43x or

O O

1x

Isolate the absolute value on one side of the inequality sign. 43x

or 1x 7x or3 3 3 3

5 5 GO

Example 5 Solve. Then graph the solution.

512x

Absolute value cannot be less than zero (cannot be negative)! Thus there are no values that will be less than negative one.

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Example 6 Solve. Then graph the solution. 849x32

12 9x32 2 2

/

69x3 69x3

3x315

1x5

–5 –1• •

4 4

99

3 3

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96

3

Example 7 Solve. Then graph the solution.

52

4a3

52

4a3

104a310

2a314

2• •

2 2

3 3

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522

4a35

446a314

5

314

4

3

Example 8 Solve. Then graph the solution.

76x

Absolute value will be zero or greater (it cannot be negative). Thus x can be any real number and the absolute value will be greater than negative one.

all real numbers

0•

Example 9 Solve and then graph.

1

GO

44

81x3 81x3

84x

84xor84x

4x 14x

4412x

1 112x or

12• •4

When solving an absolute-value inequality:

The less than symbol represents the “AND” type of inequality. It graphs as a line segment and has less (fewer) solutions.

The greater than symbol represents the “OR” type of inequality. It graphs as two opposite rays and has a greater number of solutions.

GO LA

6-A12 Pages 332-333 # 8–16,23–26,46-51.