Example: Solve |3x + 1| = |2x + 4| 3x + 1 = 2x + 4 OR 3x + 1 = -(2x + 4) x + 1 =4 3x + 1 = -2x 4 – x = 3 5x = -5 x = -1 → Solution: {-1, 3} Check: |3(-1) + 1| = |2(-1) + 4| = |2| →|-2| or |3(3) + 1| = |2(3) + |10| = |10| Yes! 4| → Practice: 1) |x 1| = |5x + 7| 2) |z + 6| = |2z 3| – – 3) |x + 2| = |x 5| – Special Cases with Absolute Value Note: Absolute value can never be negative, and is equal to zero only when the expression in the absolute value is equal to zero. Examples: |2x + 7| = -3 Impossible! Therefore, No Solution |4z 10| < -13 Impossible! Therefore, No Solution – |5w 3| = 0 Only If 5w 3 = 0. Therefore w = 3/5 – – |3d 2 | > -1 This will always be true! Therefore, All Real Numbers – |x + 2| 0 Need only solve for = 0. Therefore x = -2 ≤ Practice: 1) |5x 2| 0 2) |8n + 4| -4 – ≥ ≤
Example: Solve |3x + 1| = |2x + 4| 3x + 1 = 2x + 4 OR 3x + 1 = -(2x + 4) x + 1 =4 3x + 1 = -2x – 4 x = 3 5x = -5 → x = -1 Solution: {-1, 3} Check: |3(-1) + 1| = |2(-1) + 4| →|-2| = |2| or |3(3) + 1| = |2(3) + 4| → |10| = |10| Yes! Practice: 1) |x – 1| = |5x + 7| 2) |z + 6| = |2z – 3| 3) |x + 2| = |x – 5| Special Cases with Absolute Value Note: Absolute value can never be negative, and is equal to zero only when the expression in the absolute value is equal to zero. Examples: |2x + 7| = -3 Impossible! Therefore, No Solution |4z – 10| < -13 Impossible! Therefore, No Solution |5w – 3| = 0 Only If 5w – 3 = 0. Therefore w = 3/5 |3d – 2 | > -1 This will always be true! Therefore, All Real Numbers |x + 2| ≤ 0 Need only solve for = 0. Therefore x = -2 Practice: 1) |5x – 2| ≥ 0 2) |8n + 4| ≤ -4