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Solving Open Sentences Involving Absolute Value. | | | | | | | | | |. | | | | | | | | | |. – 3 – 2 – 1 0 1 2 3 4 5 6. – 5 – 4 – 3 – 2 – 1 0 1 2 3 4. Math Pacing. Solving Open Sentences Involving Absolute Value. There are three types of open sentences that can involve absolute value.

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math pacing

Solving Open Sentences Involving Absolute Value

| | | | | | | | | |

| | | | | | | | | |

– 3 – 2 – 1 0 1 2 3 4 5 6

– 5 – 4 – 3 – 2 – 1 0 1 2 3 4

Math Pacing
solving open sentences involving absolute value

Solving Open Sentences Involving Absolute Value

There are three types of open sentences that can involve absolute value.

Solving Open Sentences Involving Absolute Value

Consider the case | x | = n.

| x | = 5 means the distance between 0 and x is 5 units

If | x | = 5, then x = – 5 or x = 5.

The solution set is {– 5, 5}.

solving open sentences involving absolute value3

Solving Open Sentences Involving Absolute Value

When solving equations that involve absolute value, there are two cases to consider:

Solving Open Sentences Involving Absolute Value

Case 1 The value inside the absolute value symbols is positive.

Case 2 The value inside the absolute value symbols is negative.

Equations involving absolute value can be solved by graphing them on a number line or by writing them as a compound sentence and solving it.

example 5 1a

means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction.

Answer: The solution set is

Solve an Absolute Value Equation

Example 5-1a

Method 1 Graphing

The distance from –6 to –11 is 5 units.

The distance from –6 to –1 is 5 units.

example 5 1a5

Write as or

Original inequality

Case 1

Case 2

Subtract 6 from eachside.

Simplify.

Answer: The solution set is

Solve an Absolute Value Equation

Example 5-1a

Method 2 Compound Sentence

example 5 2a

So, an equation is .

Write an Absolute Value Equation

Example 5-2a

Write an equation involving the absolute value for the graph.

Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1.

The distance from 1 to –4 is 5 units.

The distance from 1 to 6 is 5 units.

example 5 2a8

Check Substitute –4 and 6 into

Answer:

Write an Absolute Value Equation

Example 5-2a
example 5 2b

Answer:

Write an Absolute Value Equation

Example 5-2b

Write an equation involving the absolute value for the graph.

solving open sentences involving absolute value10

Solving Open Sentences Involving Absolute Value

Consider the case | x | < n.

| x | < 5 means the distance between 0 and x is LESS than 5 units

Solving Open Sentences Involving Absolute Value

If | x | < 5, then x > – 5 andx < 5.

The solution set is {x| – 5 < x < 5}.

solving open sentences involving absolute value11

Solving Open Sentences Involving Absolute Value

When solving equations of the form | x | < n, find the intersection of these two cases.

Solving Open Sentences Involving Absolute Value

Case 1 The value inside the absolute value symbols is less than the positive value of n.

Case 2 The value inside the absolute value symbols is greater than negative value of n.

example 5 3a

Then graph the solution set.

Write as and

Case 2

Case 1

Original inequality

Add 3 to each side.

Simplify.

Answer: The solution set is

Solve an Absolute Value Inequality (<)

Example 5-3a
example 5 3b

Then graph the solution set.

Answer:

Solve an Absolute Value Inequality (<)

Example 5-3b
solving open sentences involving absolute value14

Solving Open Sentences Involving Absolute Value

Consider the case | x | > n.

| x | > 5 means the distance between 0 and x is GREATER than 5 units

Solving Open Sentences Involving Absolute Value

If | x | > 5, then x < – 5 orx > 5.

The solution set is {x| x < – 5 or x > 5}.

solving open sentences involving absolute value15

Solving Open Sentences Involving Absolute Value

When solving equations of the form | x | > n, find the union of these two cases.

Solving Open Sentences Involving Absolute Value

Case 1 The value inside the absolute value symbols is greater than the positive value of n.

Case 2 The value inside the absolute value symbols is less than negative value of n.

example 5 4a

Then graph the solution set.

Write as or

Case 2

Case 1

Original inequality

Add 3 to each side.

Simplify.

Divide each side by 3.

Simplify.

Solve an Absolute Value Inequality (>)

Example 5-4a
example 5 4a17

Answer: The solution set is

Solve an Absolute Value Inequality (>)

Example 5-4a
example 5 4b

Then graph the solution set.

Answer:

Solve an Absolute Value Inequality (>)

Example 5-4b
solving open sentences involving absolute value19

Solving Open Sentences Involving Absolute Value

In general, there are three rules to remember when solving equations and inequalities involving absolute value:

Solving Open Sentences Involving Absolute Value

  • If then or(solution set of two numbers)
  • If then and(intersection of inequalities)
  • If then or(union of inequalities)