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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 only works for a ≥ 0
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?
Absolute Value Example
|a| = a if a ≥ 0
if a < 0|a| = -a
Evaluate Absolute Value
Evaluate |5-x2| for x = -3
|5-x2|
|5-(-3)2|
|5-9|
|-4|
4
Substitute using x = -3
Simplify the exponent
Subtraction
Definition of Absolute Value
Given
Evaluate Absolute Value
Evaluate |x2-4x-6| for x = -1
|x2-4x-6|
|(-1)2-4(-1)-6|
|1-4(-1)-6|
|-1|
1
Substitute using x = -1
Simplify the exponent
Subtraction
Definition of Absolute Value
Given
|5-6| Addition|1+4-6| Multiplication
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
Case 2:
x -1 = -5
x = -4
x = {-4, 6}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 1:
2x -3 = 16
x = 19 2
Case 2:
2x -3 = -16
2x = 19 2x = -13
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.
x = { }This means the 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
5 = x
Case 2:
10 = 2x
2x +8 = -(4x - 2)
6x = -6
2x +8 = -4x +2
x = -1
x = {-1,5}
Why Should I Check It?
Check: |2x +8| = 4x -2; x = {-1,5}
Check 5:
|2(5) +8| = 4(5) -2
|18| = 18
Check -1:
|10 +8| = 20 -2
18 = 18
|2(-1) +8| = 4(-1) -2
|6| = -6|-2 +8| = -4 -2
6 = -6
Good answer. We’ll keep you.
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.
x = 5 This is right.
-1 didn’t check, so it is rejected!
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.