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1-5 Absolute Value Equations and Inequalities

1-5 Absolute Value Equations and Inequalities. Algebraic Definition of Absolute Value - If x > 0, then | x | = x - If x < 0. then | x | = -x. –3 x = –9 –3 x = –21 Subtract 15 from each side of both equations. Check: |15 – 3 x | = 6 |15 – 3 x | = 6

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1-5 Absolute Value Equations and Inequalities

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  1. 1-5 Absolute Value Equations and Inequalities

  2. Algebraic Definition of Absolute Value - If x > 0, then | x | = x - If x < 0. then | x | = -x

  3. –3x = –9 –3x = –21 Subtract 15 from each side of both equations. Check: |15 – 3x| = 6 |15 – 3x| = 6 |15 – 3(3)| 6 |15 – 3(7)| 6 |6| 6 |–6| 6 6 = 6 6 = 6 Absolute Value Equations and Inequalities Lesson 1-5 Additional Examples Solve |15 – 3x| = 6. |15 – 3x| = 6 15 – 3x = 6 or 15 – 3x = –6 The value of 15 – 3x can be 6 or –6 since |6| and |–6| both equal 6. x = 3 or x = 7 Divide each side of both equations by –3.

  4. You try: | 2y - 6 | = 12

  5. Extraneous Solution: a solution of an equation derived from an original equation that is not a solution to the original equation. Extraneous Solution Video

  6. 9 2 |x + 9| = Divide each side by –2. 9 2 9 2 x + 9 = or x + 9 =– Rewrite as two equations. Check: 4 – 2 |x + 9| = –5 4 – 2|x + 9| = –5 4 – 2 |–4.5 + 9| –5 4 – 2 |–13.5 + 9| –5 4 – 2 |4.5| –5 4 – 2 |–4.5| –5 4 – 2 (4.5) –5 4 – 2 (4.5) –5 –5 = –5 –5 = –5 Absolute Value Equations and Inequalities Lesson 1-5 Additional Examples Solve 4 – 2|x + 9| = –5. 4 – 2|x + 9| = –5 –2|x + 9| = –9 Add –4 to each side. x = –4.5 or x =–13.5 Subtract 9 from each side of both equations.

  7. 3x – 4 = –4x – 1 or 3x – 4 = –(–4x –1) Rewrite as two equations. 7x – 4 = –1 3x – 4 = 4x + 1 Solve each equation. 7x = 3 – x = 5 3 7 x = or x = –5 Absolute Value Equations and Inequalities Lesson 1-5 Additional Examples Solve |3x – 4| = –4x – 1. |3x – 4| = –4x – 1

  8. = / 3 7 3 7 The only solution is –5. is an extraneous solution. –4() – 1 Absolute Value Equations and Inequalities Lesson 1-5 Additional Examples (continued) Check: |3x – 4| = –4x – 1 |3x – 4| = –4x – 1 |3( ) – 4| |3(–5) – 4| (–4(–5) –1) 3 7 19 7 19 7 |– | – |–19| 19 19 7 19 7 19 = 19

  9. 2x < 2 2x > 8 Solve for x. Absolute Value Equations and Inequalities Lesson 1-5 Additional Examples Solve |2x – 5| > 3. Graph the solution. |2x – 5| > 3 2x – 5 < –3 or 2x – 5 > 3 Rewrite as a compound inequality. x < 1 or x > 4

  10. < < < < < – – – – – > > > – – – –2|x + 1| + 5 –3 –2|x + 1| –8 Isolate the absolute value expression. Subtract 5 from each side. |x + 1| 4 Divide each side by –2 and reverse the inequality. –4 x + 1 4 Rewrite as a compound inequality. –5 x 3 Solve for x. Absolute Value Equations and Inequalities Lesson 1-5 Additional Examples Solve –2|x + 1| + 5 –3. Graph the solution.

  11. Homework pg 36 3-27 every 3rd

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