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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 1 by Gregory Hauca Glencoe adapted from presentations by Linda Stamper

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Sentences have verbs and phrases do not. Phrases Phrases are translated into expressions. ex. 2x + 3 Sentences Sentences 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 = 8, An inequality has an inequality sign between them. ex > 0, 3x + 1 = 4 2x + 1 < 4x - 3

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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 True > > 10 False 13 = 13 5 > Open sentences are neither true nor false until the variable(s) have been replaced by specific values and the open sentence simplified.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 = 5 True (2) + 3 = 5

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

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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 solution The value or values of the variable that make an equation true is the solution. Solving Equations and Inequalities using Replacement Sets

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Ex.1 solution The value or values of the variable that make an equation true is the solution. Find the solution set for 5(x + 2) = 40 if the replacement set is {4, 5, 6, 7} solution: n = 6

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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 Whats 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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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

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

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solution The value or values of the variable that make an inequality true is the solution. Ex.2 Find the solution set for 9 > 2y 5 if the replacement set is {5, 6, 7, 8} solution: y = {5, 6}

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

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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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HW1-A6 Pages #14-30 even,34,36,53,57, Page 9 # Remember! Correct your odd-numbered homework problems in the back of the book!

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