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.