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Understanding Absolute Value Equations and Inequalities: Congruency, Rules, and Solutions

This week, we focus on absolute value equations and inequalities, highlighting their properties, solution strategies, and congruency statements. We explore various scenarios where absolute value equations can yield two answers and apply rules for solving inequalities. Key examples demonstrate how to determine solutions based on directionality of inequalities and effects of flipping symbols. The assignment encourages practicing found concepts through textbook exercises emphasizing absolute values in algebraic contexts.

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Understanding Absolute Value Equations and Inequalities: Congruency, Rules, and Solutions

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  1. Week 4 Warm Up 11.10.11 Q N Is ∆NQM ≅ ∆PMQ? Give congruency statements to prove it. M P

  2. Absolute value equations have 2 answers. Rule 1 x = 5 5 -5 x = -5 x | | = 5 Ex 1 | 5 | = 5 | -5 | = 5 5 = 5 5 = 5 x = -5 and 5

  3. Ex 2 -3 ≤ x < 7 x is greater than or equal to -3 and less than 7 When arrows point toward each other: Rule 2 AND

  4. x < -5 or x ≥ 6 Ex 3 x is less than -5 or greater than or equal to 6 When arrows point away from each other: Rule 3 OR

  5. Write two equations, one with a positive and one with a negative answer. Rule 4 | x - 2 | = 5 | x - 2 | = 5 Ex 4 x - 2 = -5 x - 2 = 5 x = 5 + 2 x = -5 + 2 x = x = -3 7 | x - 2 | = 5 | -3 - 2 | = 5 | 7 - 2 | = 5 | - 5 | = 5 | 5 | = 5 5 = 5 5 = 5 x = -3, 7

  6. Flip the symbol and change the right side to the opposite for second inequality. Rule 5 Ex 5 | 4x - 8 | ≤ 24 4x - 8 ≤ 24 4x - 8 ≥ -24 4x ≥ -24 + 8 4x ≤ 24 + 8 4x ≤ 32 4x ≥ -16 x ≤ 8 x ≥ -4 -4 ≤ x ≤ 8

  7. Ex 6 | -2x + 12 | > 6 -2x + 12 > 6 -2x + 12 < -6 -2x < -6 - 12 -2x > 6 - 12 -2x > -6 -2x < -18 x < 3 x > 9 x < 3 or x > 9

  8. Ex 7 | -7x + 21 | < -9 Absolute value problems cannot be equal or unequal to a negative number. Rule 6 no solution

  9. ______ _____ equations have 2 answers. Review Do: 1 | 6x - 9 | ≤ 27 Assignment: Textbook Page 259, 43 - 59 odds.

  10. Ex 5 | 3x + 6 | = 21 3x + 6 = 21 3x + 6 = -21 3x = -21 - 6 3x = 21 - 6 3x = 15 3x = -27 x = 5 x = -9 x = -9, 5

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