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.