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1.4 Absolute Value Equations Absolute value: The distance to zero on the number line. We use two short, vertical lines so that |x| means “the absolute value of x” Absolute Value Example |3| = 3 |-3| = 3 |a| = a Absolute Value Example |a| = a This won’t work all of the time. Does it work for a = 1? Does it work for a = 0? Does it work for a = -1? This only works for a ≥ 0 Absolute Value Example |a| = a if a ≥ 0 |a| = -a if a < 0 Evaluate Absolute Value Evaluate |5-x2| for x = -3 |5-x2| Given |5-(-3)2| Substitute using x = -3 |5-9| Simplify the exponent |-4| Subtraction Definition of Absolute Value 4 Evaluate Absolute Value Evaluate |x2-4x-6| for x = -1 |x2-4x-6| |(-1)2-4(-1)-6| |1-4(-1)-6| |1+4-6| |5-6| |-1| 1 Given Substitute using x = -1 Simplify the exponent Multiplication Addition Subtraction Definition of Absolute Value Solving Absolute Value Equations Your biggest concern with solving is that there are typically 2 cases to solve! Solve: |x -1| = 5 For x -1 being positive, we can just throw the ||’s into the trash and continue. But what about the case where x -1 is negative? Solving Absolute Value Equations Solve: |x -1| = 5 Case 1: x -1 = 5 x=6 x = {-4, 6} Case 2: x -1 = -5 x = -4 There’s more than one answer. That means there is a set of answers. So we need to use { }’s around our set. Solving Absolute Value Equations Solve: |2x -3| = 16 Case 2: Case 1: 2x -3 = -16 2x -3 = 16 2x = 19 2x = -13 x = 19 2 x = -13 2 { x = -13 , 19 2 2 } Oops! Solving the impossible? Solve: |2x -3| +5 = 0 |2x -3| = -5 |something| is trying to be negative ??? Solving the impossible? Solve: |2x -3| +5 = 0 So, no, this problem doesn’t have a solution. This means the x={} solution set is empty. x=∅ Same thing, except fancier. Why Should I Check It? So why do math teachers make such a big deal about checking your answers? Isn’t being careful while solving good enough? Sorry, no. Prepare to meet a most deceptive type of problem. Why Should I Check It? Solve: |2x +8| = 4x -2 Case 1: 2x +8 = 4x - 2 10 = 2x 5=x x = {-1,5} Case 2: 2x +8 = -(4x - 2) 2x +8 = -4x +2 6x = -6 x = -1 Why Should I Check It? Check: |2x +8| = 4x -2; x = {-1,5} Check -1: Check 5: |2(5) +8| = 4(5) -2 |10 +8| = 20 -2 |18| = 18 18 = 18 Good answer. We’ll keep you. |2(-1) +8| = 4(-1) -2 |-2 +8| = -4 -2 |6| = -6 6 = -6 Aaaargh! That’s a bad answer! Why Should I Check It? Edit: |2x +8| = 4x -2; x = {-1,5} x = {-1,5} This is wrong. -1 didn’t check, so it is rejected! x=5 This is right. Why Should I Check It? After you finish tonight’s homework, for every equation that you didn’t check, mark it wrong so we can save time grading.