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Warm-Up Evaluate each expression, given that x=3 and y=-2. a. |2x -9| Answer: 1) -32) 33) 154) -15 b. |y –x| Answer: 1) -52) 13) -14) 5 Solve. |3x + 6| = 9 Answer: 1) x=1, -52) x= -1, 53) x= 3, -154) x= -3, 15

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6.4: Absolute Values and Inequalities Objective : Learn how to solve absolute value inequalities.

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Review Why is the absolute value of a number always greater than or equal to zero? Two or more inequalities connected by the words _______ or _________ are a compound inequality.

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Conjunction: |a x + b| < c Means: x is between + c -c < a x +b < c Less Than when an absolute value is on the left and the inequality symbol is < or ≤, the compound sentence uses and.

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Disjunction: |a x +b| > c Means: not between! a x + b c Greater Than when an absolute value is on the left and the inequality symbol is > or ≥, the compound sentence uses or.

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Solving absolute inequalities and graphing: | x - 4| < 3 (less than is between) Means: -3 < x- 4 < 3 (solve) Graph: +4 1< x< 7 0 1 2 3 4 5 6 7 8 9

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Solving absolute inequalities and graphing: |s – 3| ≤ 12 (less than is between) Means: -12 ≤ s – 3 ≤ 12 (solve) + 3 + 3 + 3 - 9 ≤ s ≤ 15 Graph: -9 -6 -3 0 3 6 9 12 15 18 21 24

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Check Your Progress Solve each absolute value inequalities then graph. A. |y + 4| < 5 B. |z – 3| ≤ 2

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Solve and graph: | x + 9 |> 13(disjunction) Means: x + 9 13 -9 x < -22 x > 4 Graph: -25 -20 -15 -10 -5 0 5 10

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Check Your Progress Solve each absolute value inequalities and graph. A. | 3y – 3| > 9 B. |2x + 7| ≥ 11

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Change the graph to an absolute value inequality: 1. Write the inequality. (x is between) 2 < x < 8 2.Find half way between 2 and 8 It ’ s 5 (this is the median) To find the median, add the two numbers and then divide by 2. 2+8 = 5 0 1 2 3 4 5 6 7 8 9 10 2

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3. Now rewrite the inequality and subtract 5 (the median) from each section. 2 - 5 < x - 5 < 8 - 5 Combine like terms or numbers and you get-3 < x - 5 < 3 4. Write your absolute inequality | x - 5| < 3 Notice: The median is 3 units away from either number.

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Write the inequality for this disjunction: -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 1.x 4 (find the median) 2. x + 1 5 3. | x +1|>5 +1 (subtract -1 from both sides, so add 1) +1 (write x + 1 inside the absolute brackets and 5 outside positive)

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Check Your Progress Write an absolute value inequality for the graph shown -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

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Closing the lesson: Summarize the major points of the lesson and answer the Essential Question: How are absolute value inequalities like linear inequalities?

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Homework: Textbook page 316 #8-30 even, 31 – 36, 38 –40

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