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Section 1Section 1--3: Open Sentences3: Open SentencesAlgebra 1 (H)

Monday, August 9, 2010

HCPS IIIHCPS III

Standard 10: Patterns, Functions, and Algebra: SYMBOLIC REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical situation.

◦ Benchmark MA.A1.10.3: Justify the steps used in simplifying expressions and solving equations and inequalities

Solving EquationsSolving Equations

Vocabulary◦ Open Sentence: a math statement with

one or more variables.

◦ Solving the open sentence: find a value for a variable that results in a true sentence.

◦ Equation: a statement with an (=) between two expressions.

Solving EquationsSolving Equations Vocabulary (con’t)◦ Replacement Set: a group of numbers (#’s) to

pick a solution from.

◦ Set: a collection of #’s defined by braces { } and named with a capital letter.named with a capital letter.

◦ Element: a # in a set.

◦ Solution Set: set of #’s that make a statement true, written in { }.

Set A {1, 2, 3, 4, 5, 6}

Element

Example 1: Example 1: Using a Replacement Set to Solve an EquationUsing a Replacement Set to Solve an Equation

1. Find the solution set for each equation if the replacement set is {2, 3, 4, 5, 6}.

a) 4a + 7 = 23n 4a + 7 = 23 True/False

The solution set is {4}

n 4a + 7 23 True/False2 4 (2) + 7 = 8 + 7 = 15 15 23 False

3 4 (3) + 7 = 12 + 7 = 19 15 23 False

4 4 (4) + 7 = 16 + 7 = 23 23 = 23 True

5 4 (5) + 7 = 20 + 7 =27 27 23 False

6 4 (6) + 7 = 24 + 7 = 31 31 23 False

Example 1: Example 1: Con’tCon’t

Replacement set is {2, 3, 4, 5, 6}.

b) 3(8 - b) = 6

n 3 (8 – b) = 6 True/False2 3 ( 8 – 2) = 3 (6) = 18 18 6 False

The solution set is {6}

2 3 ( 8 – 2) = 3 (6) = 18 18 6 False

3 3 ( 8 – 3) = 3 (5) = 15 15 6 False

4 3 ( 8 – 4) = 3 (4) = 12 12 6 False

5 3 ( 8 – 5) = 3 (3) = 9 9 6 False

6 3 ( 8 –6) = 3 (2) = 6 6 = 6 True

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Example 2: Example 2: Use Order of Operations to Solve an EquationUse Order of Operations to Solve an Equation

2. Solve

Solving InequalitiesSolving Inequalities

Inequality: an open sentence containing <, <, >, >

Example 3: Example 3: Find the Solution Set of an InequalityFind the Solution Set of an Inequality3. Find the solution set for z + 11 > 32, if the

replacement set is {20, 21, 22, 23, 24}

z z + 11 > 32 True/False

The solution set is {21, 22, 23, 24}

20 20 + 11 = 31 31 > 32 False

21 21 + 11 = 32 32 > 32 True

22 22 + 11 = 33 33 > 32 True

23 23 + 11 = 34 34 > 32 True

24 24 + 11 = 35 35 > 32 True

Example 4: Solve an InequalityExample 4: Solve an Inequality

4. OUTDOORS. A four-wheel-drive tour of Canyon de ChellyNational Monument in Arizona costs $45 for the first vehicle and $15 for each additional vehicle. How many vehicles can the Smith family take on the tour if they want to spend no more than $100?

EXPLORE: The Smith family can spend no more than $100. So the situation can be represented as:

45 + 15x < 100

PLAN: Since no replacement set is given I will use {1, 2, 3, 4, 5}.

Example 4: Example 4: Con’tCon’tSOLVE:

x 45 + 15x < 100 True/False1 45 + 15(1) = 60 60 < 100 True

2 45 + 15(2) = 45 + 30= 75 75 < 100 True

3 45 + 15(3) 45 + 45 90 90 < 100 T

EXAMINE: The solution set is {1, 2, 3}, which means that the Smith family can take up to three additional vehicles. Therefore, the Smith family can take up to four vehicles (the first vehicle and the three additional vehicles)

3 45 + 15(3) = 45 + 45= 90 90 < 100 True

4 45 + 15(4) = 45 + 60= 105 105 < 105 False

5 45 + 15(5) = 45 + 75= 120 120 < 120 False