1-3 Open Sentences

In this section we are going to define a mathematical

sentence and the algebraic term solution.

We will also solve equations and

inequalities using replacement sets and order of operations.

Algebra 1 by Gregory HaucaGlencoe adapted from presentations by Linda Stamper

Sentences have verbs and phrases do not.

PhrasesPhrases are translated into expressions. ex. 2x + 3

SentencesSentences are translated into equations and inequalities.

In section 1-1 we discussed the difference between phrases and sentences in English.

In algebra

An equation is a mathematical sentence with the equal sign between two expressions.

ex. 5 + 3 = 8,

An inequality has an inequality sign between them.ex. 3 - 2 > 0,

3x + 1 = 4

2x + 1 < 4x - 3

An equation or inequality is “open” if it contains a variable expression.ex. 7x – 2 =

8, 3(x + 2) < 8

Equations and inequalities can be either TRUE or FALSE

Examples: 2(5) + 3 = 13 10 + 3 =

13True

7 - 2 > 8 + 2

5 > 10False

13 = 13

5 > 8 + 2

Open sentences are neither true nor false until the variable(s) have been replaced by specific values and the open sentence simplified.

Example: x + 3 = 5

5 = 5True

(2) + 3 = 5

The value or values of the variable that make an equation or inequality true is called the solutionsolution.

The process of finding a value for the variable that results in a true statement is called solving.

This replacement value is called a solution. An equation or inequality may have one, many, or no solutions.

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

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

solution: n = 5

6n 7 37

6n 7 37

3743

37736

37766

377n6

3749

37742

37776

377n6

6 4 7 37

24 7 37 31 37

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

Solving Equations and Inequalities using Replacement Sets

6 5 7 37 30 7 37

37 37

Ex.1

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

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

solution: n = 6

5 x 2 40

5 x 2 40

5 5 2 40

5 7 40

35 40

5 x 2 40

5 6 2 40

5 8 40

40 40

5 x 2 40

5 7 2 40

5 9 40

45 40

5 4 2 40

5 6 4030 40

Inequalities.

< is less than 4 < 8

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

> is greater than or equal to 20 > 20

inequality symbol meaning example

What’s the importance of is in each of the above?

It is a verb. Without it you will have a subtraction or addition not an inequality.

n is less than 4

Learn to distinguish inequality from subtraction.

n less than 4

Expression (no verb)

Is this an expression or a

sentence?

4−n

Is this an expression or a

sentence?

Sentence (verb is)n < 4

To solve an inequality given a replacement set, substitute each 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 an inequality true is the solutionsolution.

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

solution: n = {7, 8}

30 n 37

3736

37630

37n30

3737

37730

37n30

3738

37830

37n30

30 5 37 35 37

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

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

solution: y = {5, 6}

9 2y 5

79

5129

5629

5y29

99

5149

5729

5y29

119

5169

5829

5y29

9 2 5 5 9 10 5 9 5

Write the problem.Follow rules for order of operations.

Solving equations by applying the order of operations.

453

4213m

13 8m

21m

3

m 7

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

Solve.

3 1

Whiteboard Practice

129

y2

81y

3

y 27

1243

573x

2

9 2

x3 4 3

7

x12 3

7x

15

k3518

2853

3

5 10k

18 2

5 10k

18 8

50

k10

5 k

424

3425x 3

2

5 4 4 3x

4 8 4

20 4 3x

4 4

20 12x

16

32x

16

x 2

Find the solution set for 5x + 10 = 40 if the replacement set is {4, 5, 6, 7}

Solution: x = 6

5x 10 40

5x 10 40

5 5 10 40

25 10 40

35 40

5x 10 40

5 6 10 40

30 10 40

40 40

5x 10 40

5 7 10 40

35 10 40

45 40

5 4 10 40

20 10 40

30 40

HW1-A6 Pages 18-20 #14-30 even,34,36,53,57,64-67. Page 9 #49-54.

Remember! Correct your odd-numbered

homework problems in the back of the

book!