Absolute Value Notes

For those of you who missed it...

Absolute Value!!!

1. 2. 3.

4. 5.

6.

Domain: (-∞,∞)Range: [2,∞)

Domain: (-∞,∞)Range: (-∞, -2]

Domain: (-∞,∞)

Range: [-1,∞)

Domain: (-∞,∞)

Range: [-2,∞)

Domain: (-∞,∞)

Range: (-∞, 3]Domain: (-∞,∞)

Range: (-∞, 1]

Answers to Absolute Value Worksheet

f(x) = 2|x - 3| + 3f(x) = 1/3|x + 5| + 3


f(x) = -3/2|x - 5| - 4 f(x) = -3/2|x + 6| - 2


f(x) = 3|x - 4| - 10

f(x) = -2|x - 4| + 9


f(x) = 4|x + 5| + 9 f(x) = 3/5|x + 4| - 8


2x + 3 = 6 2x + 3 = -6 2x = 3 2x = -9 x = 3/2 x = -9/2

Solving Absolute Value Equations...Absolute Value:For any real number x,

|x| ={-x, if x < 0 0, if x = 0 x, if x > 0

Recall: When solving equations, isolate the absolute value. Here are a few examples...

1. 5|2x + 3| = 30 |2x + 3| = 6

Don't forget to check!!!

5|6| = 30 5|-6| = 30

solution set: {3/2, -9/2}

example 2: -2|x + 2| + 12 = 0

-2|x + 2| = -12 |x + 2| = 6

isolate the absolute value!

x + 2 = -6 x =

-8

x + 2 = 6 x =

4-2|-6| + 12 = 0 -2|6| + 12 = 0

{4, -8}

5|3×+ 7|=-65|3x + 7|=-13

absolute value cannot be negative!!

example 3:

{}

{3}

example 4:

|2x + 12| = 7x - 3

2x + 12 = 7x - 32x + 15 = 7x 15 = 5x 3 = x

2x + 12 = -(7x - 3)2x + 12 = -7x + 39x + 12 = 3

9x = -9 x = -1

|18| = 18|10| = -10

reject!

Absolute Value InequalitiesRecall: |ax+b|=c, where c>0

ax+b=c ax+b=-c|ax+b|<c think: between "and"

-c < ax+b < c

ax+b < c and ax+b > -c

ax+b>c or ax+b<-c

why?

we will express < or ≤ as an equivalent conjunction using the word AND

|ax+b|>c think: beyond "or" we will express > or ≥ as an equivalent disjunction using the word OR

I. Less than...a) |x| < 5

x < 5 and x >-5written as

10 2 3 4 5 6 7 8 9 10-1-2

-3-4-5-6-7-8-9-10

solution set: {x: -5< x < 5}

Graph on a number line!

use open circles!

shade between!!!

b) |2x - 1| < 11

2x-1<11 and 2x-1>-11 2x < 12 and 2x > -10 x < 6 and x > -5

10 2 3 4 5 6 7 8 9 10-1-2

-3-4-5-6-7-8-9-10

{x: -5 < x < 6}

c) 4|2x + 3| - 11 ≤ 5

4|2x + 3| ≤ 16 |2x + 3| ≤ 4

2x + 3 ≤ 4 AND 2x + 3 ≥ -42x ≤ 1 AND 2x ≥ -7 x ≤ 1/2 AND x ≥ -7/2

-1 0-2

-3-4-5 1 2 3 4 5

notice closed ends!

d) |7x + 10| < 0

think....can an absolute value be negative???NO!! {}

II. Greater than...a) |x| > 5

x > 5 or x < -5written as

10 2 3 4 5 6 7 8 9 10-1-2

-3-4-5-6-7-8-9-10

solution set: {x: x > 5 or x < -5}

Interval notation (we will not use this, just set, but as an FYI): (-∞, -5) ∪ (5, ∞)

Graph on a number line! use open circles!

shade beyond!!!

b) |2x - 1| > 11

2x-1>11 or 2x-1<-11 2x > 12 or 2x < -10 x > 6 or x < -5

10 2 3 4 5 6 7 8 9 10-1-2

-3-4-5-6-7-8-9-10

{x: x > 6 or x < -5}

c) 4|2x + 3| - 11 ≥ 54|2x + 3| ≥ 16 |2x + 3| ≥ 4

2x + 3 ≥ 4 OR 2x + 3 ≤ -42x ≥ 1 OR 2x ≤ -7 x ≥ 1/2 OR x ≤ -7/2

-1 0-2

-3-4-5 1 2 3 4 5

notice closed ends!

d) |7x + 10| > 0

think....when is an absolute value greater than 0???

always!!

{x: x ∈ R }x is a real number!

-1 0-2

-3-4-5 1 2 3 4 5

LAST ONE!

5 < |x + 3| ≤ 7

|x + 3| >5 |x + 3| ≤ 7x + 3 > 5 or x + 3 < -5 x+ 3 ≤ 7 and x + 3 ≥ -7 x > 2 or x < -8 x ≤ 4 and x ≥ -10

now graph it! graph above the number line and look for the overlap. This is where your solution will appear.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

{x: -10 ≤ x < 8 or 2 < x ≤ 4}

Remember to see me, email me or ask on the wiki if you have questions!!

-Ms. P

Technology

Absolute Value Notes