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Warm-Up • Evaluate each expression, given that x=3 and y=-2. a. |2x -9| Answer: 1) -3 2) 3 3) 15 4) -15 2) 1 3) -1 4) 5 2) x= -1, 5 3) x= 3, -15 4) x= -3, 15 b. |y –x| Answer: 1) -5 • Solve. |3x + 6| = 9 Answer: 1) x=1, -5 6.4: Absolute Values and Inequalities Objective: • Learn how to solve absolute value inequalities. 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. Conjunction: |ax + b| < c Means: x is between + c -c < ax +b < c Less Than when an absolute value is on the left and the inequality symbol is < or ≤, the compound sentence uses and. Disjunction: Means: |ax +b| > c not between! ax + b < -c or ax + b > c Greater Than when an absolute value is on the left and the inequality symbol is > or ≥, the compound sentence uses or. Solving absolute inequalities and graphing: |x - 4| < 3 Means: (less than is between) -3 < x- 4 < 3 (solve) +4 +4 +4 1< x< 7 Graph: 0 1 2 3 4 5 6 7 8 9 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 Check Your Progress • Solve each absolute value inequalities then graph. • A. |y + 4| < 5 • B. |z – 3| ≤ 2 Solve and graph: |x + 9 |> 13 (disjunction) Means: x + 9 < -13 -9 -9 or x + 9 > 13 x < -22 Graph: -25 -20 -15 -10 -5 0 5 10 -9 -9 x>4 Check Your Progress • Solve each absolute value inequalities and graph. • A. | 3y – 3| > 9 • B. |2x + 7| ≥ 11 Change the graph to an absolute value inequality: 0 1 2 3 4 5 6 7 8 9 10 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 2 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. Write the inequality for this disjunction: -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 1. x < -6 +1 +1 2. x + 1 < - 5 3. |x+1|>5 or x>4 +1 +1 (find the median) (subtract -1 from both sides, so add 1) x+1 > 5 (write x + 1 inside the absolute brackets and 5 outside positive) 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 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Closing the lesson: • Summarize the major points of the lesson and answer the Essential Question: How are absolute value inequalities like linear inequalities? Homework: • Textbook page 316 #8-30 even, 31 – 36, 38 –40