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3.6 Abs Value Equations Hornick Objectives 3.6 Solve equations in one variable that contain absolute-value expressions. Remember... Recall that the absolute-value of a number is that number’s distance from zero on a number line. For example, |–5| = 5 and |5| = 5. 5 units 6 5 4 3 2 1 5 units 0 1 2 3 4 5 6 Isolate absolute value first! Then you must consider two cases… Example 1 Solve the equation. |x| = 12 |x| = 12 Think: What numbers are 12 units from 0? • 12 units 12 10 8 6 4 2 Case 1 x = 12 Case 2 x = –12 • 0 12 units 2 4 6 • 8 10 12 Rewrite the equation as two cases. The solutions are {12, –12}. You must use { } for your answer! Example 2 Solve the equation. 3|x + 7| = 24 |x + 7| = 8 Case 1 Case 2 x + 7 = 8 x + 7 = –8 – 7 –7 –7 –7 x =1 x = –15 The solutions are {1, –15}. Not all absolute-value equations have two solutions. • If the absolute-value expression equals zero, there is one solution. • If an equation states that an absolute-value is negative, there are no solutions. Example 3 Solve the equation. 8 = |x + 2| 8 8 = |x + 2| 8 +8 +8 0 = |x + 2| 0= x+2 2 2 2 = x The solution is {2}. There is only one case! Example 4 Solve the equation. 3 + |x + 4| = 0 3 + |x + 4| = 0 3 3 |x + 4| = 3 Absolute value cannot be negative! You can’t have a negative distance! This equation has no solution. You can write to show that this set is empty. Remember! Absolute value must be nonnegative because it represents a distance. Grab a whiteboard, marker, and eraser per pair! Solve the following equation. Show all work!! 3 = |x + 4| 8 Solve the following equation. Show all work!! 16 = 5|x | 4 Solve the following equation. Show all work!! 2|7x | = 14 Solve the following equation. Show all work!! 2|x+3 | = 8 Solve the following equation. Show all work!! 1.2|5x| = 3.6 Solve the following equation. Show all work!! 2|x| = 3 Solve the following equation. Show all work!! 2|5x| = 3 Challenge Solve the following equation. Show all work!! |x+4| = 3x Classwork/Homework 3.6 Practice Worksheet Exit Card Solve each equation. 1. 15 = |x| 3. |x + 1|– 9 = –9 5. 7 + |x – 8| = 6 2. 2|x – 7| = 14 4. |5 + x| – 3 = –2 Exit Card Solve each equation. 1. 15 = |x| {–15, 15} 2. 2|x – 7| = 14 {0, 14} 3. |x + 1|– 9 = –9 {–1}4. |5 + x| – 3 = –2 {–6, –4} 5. 7 + |x – 8| = 6 no solution

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