From here to there and back again Counts steps Measures distance.

Absolute Value

From here to there and back again Counts steps Measures distance

1. Additive inverse - the opposite of a number. Two numbers are additive inverses if their sum is zero

2. Absolute value of x: lxl = 3. Absolute value is non-negative because it is the distance from zero to x.

Example for x = l-3l

Vocabulary

2x

I am going to my friend’s

house.

Absolute Value: Counts the Steps

One


One, two


One, two, three


One, two, three, four


One, two, three, four,

five



five, six



five, six, seven



five, six, seven, eight



five, six, seven, eight,

nine

Absolute Value: Counts the Steps9 steps to go from left to right

It’s time to go home.

How many steps to my house?


One


One, two


One, two, three


One, two, three, four


One, two, three, four, five


One, two, three, four, five, six


One, two, three, four, five, six,

seven


One, two, three, four, five, six, seven, eight


One, two, three, four, five, six,

seven, eight, nine

Absolute Value: Counts the Steps9 steps to go from right to left

Do the steps need to be equidistant? Yes

The absolute-value of a number is that number’s distance from zero on a number line.

For example, |–9| = ____.

The absolute-value of a number is that numbers distance from zero on a number line.

For example, |–9| = 9.

Absolute Value

Absolute Value

Represents distanceCount the steps

The absolute value of a number is the distance from zero on a number line. Example: |5| read as “the absolute value of 5”Example: |-5| read as “the absolute value of 5”

5 units 5 units

210123456 6543- - - - - -

|5| = 5|–5| = 5

Note: |5|= l-5 l= 5

Absolute Value of x

l x l

the distance from zero to x on a number line

|x|the distance from zero on a number line to x

x units x units

0 x-

|x| = x|–x| = x

x

Additive inverse The opposite of a number. Two numbers are additive inverses if their sum is zero.

x + (-_____) = 0

Additive inverse The opposite of a number. Two numbers are additive inverses if their sum is zero.

x + (-x) = 0

Absolute value Is always a positive number except for 0 as 0 is neither positive or negative.

Absolute Value Function

Graph y = lxl

Absolute-value graphs are V-shaped.

axis of symmetry - line that divides the graph into two congruent halves

vertex is the “corner" point on the graph.

From the graph of y = |x|, you can tell that:

• the axis of symmetry is the y-axis (x = 0).

• the vertex is (0, 0).

• the domain (x-values) is the set of all real numbers.

• the range (y-values) is described by y ≥ 0.

• y = |x| is a function because each domain value has exactly one range value.

• the x-intercept and the y-intercept are both 0.

Label the axis of symmetry and the vertex. Identify the intercepts, and give the domain and range.

Example 1A: Absolute-Value Functions

y = |x| + 1

y = |x| +1

x –2 –1 0 1 2

3 2 1 2 3

Example 1A Continued

• the axis of symmetry is the y-axis (x = 0).

• the vertex is (0, 1).• there are no x-intercepts. • the y-intercept is +1.• the domain is all real numbers.• the range is described by y ≥ 1.

From the graph you can tell that Axis of symmetry

Vertex

Graph


Example 1B: Absolute-Value Functions

y = |x – 4|

y = |x –4|

x –2 0 2 4 6

6 4 2 0 2

• the axis of symmetry is x = 4.

• the vertex is (4, 0).• the x-intercept is +4. • the y-intercept is +4.• the domain is all real numbers.• the range is described by y ≥ 0.

From the graph you can tell that

Example 1B Continued

axis of symmetry

vertex

Graph


f(x) = 3|x|

f(x) =3|x|

x –2 –1 0 1 2

6 3 0 3 6

Example 1C

x

Example 1C Continued

• the axis of symmetry is x = 0.

• the vertex is (0, 0).• the x-intercept is 0. • the y-intercept is 0.• the domain is all real numbers.• the range is described by y ≥ 0.

From the graph you can tell that

Axis of symmetry

Vertex

x

Absolute ValueSolve equations in one variable that contain absolute-value expressions.

Represents ________Absolute Value

Represents distanceAbsolute Value

Represents distanceCounts ____

Absolute Value

Absolute value represents distanceAbsolute value counts steps

Absolute Value

The absolute-value of a number is that number’s distance from zero on a number line.

Example |–6| = 6

5 4 3 2 0 1 2 3 4 56 1 6

6 units

Both 6 and –6 are a distance of 6 units from 0, so both 6 and –6 have an absolute value of 6.

6 units

Given |x| = 6Solve for xThis equation asks, “What values of x have an absolute value of 6?” The solutions are 6 and –6. Notice this equation has two solutions.

Example 1A: Solving Absolute-Value Equations

Solve for x.

|x| = 12

Case 1 x = 12

Case 2 x = –12

The solutions are 12 and –12.

Think: Which numbers are 12 units from 0?

Rewrite the equation as two cases.

Check your work|x| = 12

|12| = 12

12 = 12

|x| = 12

|12| = 12

12 = 12

Example 1B: Solving Absolute-Value Equations

3|x + 7| = 24

|x + 7| = 8


Case 1 x + 7 = 8

Case 2 x + 7 = –8

– 7 = –7 – 7 = – 7 x = 1 x = –

15

Since |x + 7| is multiplied by 3, divide both sides by 3 to undo the multiplication.

Think: What numbers are 8 units from 0?

Rewrite the equations as two cases. Since 7 is added to x subtract 7 from both sides of each equation.

Example 1B Continued

3|x + 7| = 24


Check 3|x + 7| = 24 3|x + 7| = 24

3|8| = 24

24 = 24

3|1 + 7| = 24

3(8) = 24

3|15 + 7| = 24

24 = 24

3|8| = 24

3(8) = 24

Solve each equation. Check your answer.

Example 1c

|x| – 3 = 4

|x| – 3 = 4

+ 3 = +3

|x| = 7

Case 1 x = 7

Case 2 –x = 7

–1(–x) = –1(7)

x = –7x = 7


Since 3 is subtracted from |x|, add 3 to both sides.

Think: what numbers are 7 units from 0?

Rewrite the case 2 equation by multiplying by –1 to change the minus x to a positive..

Check |x| 3 = 4 |x| 3 = 4

7 3 = 4

|7| 3 = 4

4 = 4

| 7| 3 = 4

7 3 = 4

4 = 4

Solve the equation. Check your answer.

Example 1c Continued

|x| 3 = 4

The solutions are 7 and 7.


Example 1d

|x 2| = 8

|x 2| = 8 Think: what numbers are 8 units from 0?

+2 = +2

Case 1 x 2 = 8

x = 10

+2 = +2

x = 6

Case 2 x 2 = 8

Rewrite the equations as two cases. Since 2 is subtracted from x add 2 to both sides of each equation.



Example 1d Continued

|x 2| = 8


Check |x 2| = 8 |x 2| = 8

10 2| = 8

|10 2| = 8

8 = 8

| 6 + (2)| = 8

6 + 2 = 8

8 = 8

Absolute value equations most often have two solutions, but not all do:

1. If the absolute-value equals 0, there is one solution.

2. If the absolute-value is negative, there are no solutions.

Solutions to Absolute Value Equations

Example 2A


8 = |x + 2| 8

8 = |x + 2| 8+8 = + 8

0 = |x +2|

0 = x + 22 = 2

2 = x

Since 8 is subtracted from |x + 2|, add 8 to both sides to undo the subtraction.

There is only one case. Since 2 is added to x, subtract 2 from both sides to undo the addition.

Example 2A Continued


8 = |x +2| 8

8 = 8

8 = |2 + 2| 8

8 = |0| 8

8 = 0 8

To check your solution, substitute 2 for x in your original equation.

Solution is x = 2

Check8 =|x + 2| 8

Example 2B


3 + |x + 4| = 0

3 + |x + 4| = 0

3 = 3

|x + 4| = 3

Since 3 is added to |x + 4|, subtract 3 from both sides to undo the addition.

Absolute values cannot be negative.

This equation has no solution.

Remember!

Absolute value must be nonnegative because it represents distance.

Example 2c


2 |2x 5| = 7

2 |2x 5| = 7

2 = 2

|2x 5| = 5

Since 2 is added to |2x 5|, subtract 2 from both sides to undo the addition.

Absolute values cannot be negative.|2x 5| = 5

This equation has no solution.

Since |2x 5| is multiplied by a negative 1, divide both sides by negative 1.

Example d


6 + |x 4| = 6

6 + |x 4| = 6 +6 = +6

|x 4| = 0

x 4 = 0

+ 4 = +4

x = 4

Since 6 is subtracted from |x 4|, add 6 to both sides to undo the subtraction.

There is only one case. Since 4 is subtracted from x, add 4 to both sides to undo the addition.


Example 2d Continued

6 + |x 4| = 6

6 + |4 4| = 6

6 +|0| = 6

6 + 0 = 6

6 = 6

6 + |x 4| = 6

The solution is x = 4.

To check your solution, substitute 4 for x in your original equation.

From here to there and back again Counts steps Measures distance.

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