1-3 Open Sentences

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Algebra 1 Glencoe McGraw-Hill Linda Stamper

A mathematical statement with one or more variables is called an open sentence.

1510xExample An open sentence with an equal sign, =, is an equation.

1510xExample

An open sentence with the symbol <, <, >, or > is an inequality.

An open sentence is neither true nor false until the variable/s have been replaced by specific values. The process of finding a value for the variable that results in a true sentence is called solving the open sentence. This replacement value is called a solution. The value or values of the variable that

make the statement true is the solutionsolution.

A set of numbers from which replacements for a variable may be chosen is called a replacement set. A set is a collection of numbers. It is often shown using braces, { }. Each number in the set is called an element, or member. The solution set of an open sentence is the set of elements from the replacement set that makes the open sentence true.

1510x

The value or values of the variable that make the statement true is the solutionsolution.

1510x Replacement set {3, 4, 5, 6, 7}

Solution set is {5}.

Solution set is {5, 6, 7}.

To solve an equation given a replacement set, substitute eacheach value into the equation and simplify to determine if it makes the equation a true statement.

The value or values of the variable that make the statement true is the solutionsolution.

Find the solution set for 6n + 7 = 37 if the replacement set is {4, 5, 6, 7}

{5}

3731

37724

37746

377n6

3737

37730

37756

377n6

3743

37736

37766

377n6

3749

37742

37776

377n6

The value or values of the variable that make the statement true is the solutionsolution.

Example 1 Find the solution set for 5(x + 2) = 40 if the replacement set is {4, 5, 6, 7}

{6}

4030

4065

40245

402x5

4035

4075

40255

402x5

4040

4085

40265

402x5

4045

4095

40275

402x5

An inequality is a statement formed by placing an inequality symbol between two expressions.

< is less than 4 < 8

< is less than or equal to 4 < 8 > is greater than 20 > 5

> is greater than or equal to 20 > 5

inequality symbol meaning example

What’s the importance of is in each of the above?It represents the verb in the statement. Without it you will have an expression not a statement.

n is less than 4

Beware of inequality versus subtraction.

n less than 4

expression

Is this an expression or a

sentence?

4−n(translation)

Is this an expression or a

sentence?

sentence(translation)

n < 4

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To solve an inequality given a replacement set, substitute eacheach value into the inequality and simplify to determine if it makes the inequality a true statement.

The value or values of the variable that make the statement true is the solutionsolution.

Find the solution set for 30 + n > 37 if the replacement set is {5, 6, 7, 8}

{7, 8}

3735

37530

37n30

3736

37630

37n30

3737

37730

37n30

3738

37830

37n30

The value or values of the variable that make the statement true is the solutionsolution.

Example 2 Find the solution set for 9 > 2y − 5 if the replacement set is {5, 6, 7, 8}

{5, 6}

59

5109

5529

5y29

79

5129

5629

5y29

99

5149

5729

5y29

119

5169

5829

5y29

Write the problem.Follow rules for order of operations.

Solve the equation by applying the order of operations.

4534213

m

13813

321

7

Solve.

Example 3

Example 4

Example 5

129

y2

1243573

x2

k3518

2853

381

27

1243

579

343

29

31229

157

3218

105

818105

1050

5

Solve.

Example 6

424

3425x 3

2

484

3445

44

3420

161220

1632

2

32931831

34y

Solve.

3295431

34

326331

34

86331

34

86331

12

82112

89 1

Example 7

Solve.

128.088.07m 128.88.07

128.86.5

124.14

4.2

Example 8

Solve.

2314584486364212438

y

61458448

624212438

7458448

62442438

2858448

6842438

338448

634438

26444840438

18416024

184184

1

Example 9

1-A5 Pages 18-20 #22-37,53-59.