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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 by Gregory Hauca Glencoe adapted from presentations by Linda Stamper

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

Sentences have verbs and phrases do not. Phrases are translated into expressions. ex. 2x + 3 Sentences are translated into equations and inequalities. In algebra An equation is a mathematical sentence with the equal sign between two expressions. ex = 8, 3x + 1 = 4 An inequality has an inequality sign between them. ex > 0, 2x + 1 < 4x - 3

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**Equations and inequalities can be either TRUE or FALSE**

Examples: 2(5) + 3 = 13 7 - 2 > 8 + 2 = 13 5 > 8 + 2 13 = 13 5 > 10 True False An equation or inequality is “open” if it contains a variable expression. ex. 7x – 2 = 8, 3(x + 2) < 8 Open sentences are neither true nor false until the variable(s) have been replaced by specific values and the open sentence simplified.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 (2) + 3 = 5 5 = 5 True

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

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**Solving Equations and Inequalities using Replacement Sets**

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

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

Ex.1 solution: n = 6 The value or values of the variable that make an equation true is the solution.

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Inequalities. inequality symbol meaning example < is less than < 8 < is less than or equal to < 4 > is greater than > 5 > is greater than or equal to > 20 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.

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**Learn to distinguish inequality from subtraction.**

n less than 4 − 4 Expression (no verb) n is less than 4 < 4 Sentence (verb is) n Is this an expression or a sentence? Is this an expression or a sentence?

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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. Find the solution set for 30 + n > 37 if the replacement set is {5, 6, 7, 8} solution: n = {7, 8} The value or values of the variable that make an inequality true is the solution.

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

solution: y = {5, 6} The value or values of the variable that make an inequality true is the solution.

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**Solving equations by applying the order of operations.**

Solve. Write the problem. Follow rules for order of operations. The value or values of the variable that make an equation true is the solution.

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Whiteboard Practice

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

Solution: x = 6

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Homework HW1-A Pages #14-30 even,34,36,53,57, Page 9 #49-54. Remember! Correct your odd-numbered homework problems in the back of the book!

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Equations and Inequalities. Equation – A sentence stating that two qualities are equal. Includes a variable – 3x + 5 = 17 Equation variable The solution.

Equations and Inequalities. Equation – A sentence stating that two qualities are equal. Includes a variable – 3x + 5 = 17 Equation variable The solution.

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