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5.1 Addition and Subtraction Problems of Inequality Objective:
To solve and graph the solution set of an inequality by using the Addition or Subtraction Property of Inequality
Frameworks: 10.P.1, 10.P.7
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How do you read . . . a < b a is less than b a > b a is greater than b
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Inequality The open sentence x < -2 is an example of an inequality An inequality contains at least one variable and consists of 2 expressions with an inequality symbol such as <, >, or ≠ between them.
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Solving an Inequality To solve an inequality means to find a solution set. What is the solution set of x < -2? On a number line: open circle means not including this point
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Solving an Inequality How would we graph the solution of x > 1?
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Solving an Inequality The Addition and Subtraction Properties of Equality allow you to add or subtract the same number from each side of an equation to obtain an equivalent equation. x – 4 = 3 x + 2 = 5 Do inequalities work the same way?
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Equivalent Inequalities Open inequalities with the same solution set are called equivalent inequalities.
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Addition Property of Inequality For all real numbers a, b, and c, if a < b, then a + c < b + c, and if a > b, then a + c > b + c In other words, adding the same number to each side of an equality produces an equivalent equality.
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Subtraction Property of Inequality For all real numbers a, b, and c, if a < b, then a - c < b - c, and if a > b, then a - c > b - c In other words, subtracting the same number from each side of an equality produces an equivalent equality.
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After Mary paid $8.36 for a snack she had less than $2.50 left. How much money did she have originally?
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After Bill paid $7.21 at the movies, he had less than $1.75 left. How much money did he have originally?
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5.2 Multiplication & Division Problems of Inequality Objective:
To solve and graph the solution set of an inequality by using the Multiplication or Division Property of Inequality
Frameworks: 10.P.1, 10.P.7
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Solving an Inequality The Multiplication and Division Properties of Equality allow you to add or subtract the same number from each side of an equation to obtain an equivalent equation. x / 4 = 2 x * 3 = 21 Do inequalities work the same way?
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Notice: Multiplying or Dividing each side of a true equality by a negative number produces a false inequality
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Multiplication Property of Inequality, Part 1 For all real numbers a, b, and c, if a < b and c > 0, then ac < bc, and if a > b and c > 0, then ac > bc That is, multiplying each side of an inequality by the same positive number produces an equivalent inequality.
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Multiplication Property of Inequality, Part 2 For all real numbers a, b, and c, if a < b and c < 0, then ac > bc, and if a > b and c < 0, then ac < bc That is, multiplying each side of an inequality by the same negative number and reversing the order of the inequality produces an equivalent inequality.
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Division Property of Inequality, Part 1 For all real numbers a, b, and c, if a < b and c > 0, then a/c < b/c, and if a > b and c > 0, then a/c > b/c That is, dividing each side of an inequality by the same positive number produces an equivalent inequality.
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Division Property of Inequality, Part 2 For all real numbers a, b, and c, if a < b and c < 0, then ac > b/c, and if a > b and c < 0, then ac < b/c That is, dividing each side of an inequality by the same negative number and reversing the order of the inequality produces an equivalent inequality.
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Solve: 7x < -56 Divide each side by 7 x < -8 Graph:
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Solve: -⅔ x > 16 Multiply each side by the reciprocal of -⅔ Because we multiplied by a negative, change the > to a < x < -24 Graph:
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If Jill sells more than $100 worth of peanut brittle, she will win a radio. Each box of peanut brittle sells for $2.75. How many boxes must she sell to win the radio? 2.75p > 100 p > 100/2.75 p > 36.3636 Can she sell 36.36 boxes? Jill must sell 37 boxes.
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