Absolute Value and Absolute Value and GraphingGraphing

Review of Chapter 6.4Pages 295-297

What’s the Deal?What’s the Deal?

• In this lesson – We will review domain and range.– We will graph the results of how

absolute value affects variables.

y = +7y = +7

• Since every y value equals 7, we graph with a zero slope.

• y = (0)x + 7• Using an x-y box

(-4, +7)(-2, +7)( 0, +7)(+2, +7)(+4, +7)

x y-4 7-2 7 0 7+2 7+4 7

y = +7y = +7

• Arrows– Show that all points

beyond also make the equation true.

• Using an x-y box(-100, +7)(-52, +7)( 10, +7)(+20, +7)(+144, +7)

What are the domain and What are the domain and range?range?

• The domain for an equation is all the values that will work for x.

• The range for an equation is all the values that will work for y.

• Domain: {all real numbers}

• Range: {+7}

x y-4 7-2 7 0 7+2 7+4 7

7y

Number TermsNumber Terms

• Integers {…,-6,-5,-4,-3,-2,-1,0,1,2,3,4…}

• Whole Numbers { 0,1,2,3,4,5,6,7…}

• Counting Numbers { 1,2,3,4,5,6,7…}

• Real Numbers {integers, fractions, decimal numbers, repeating decimals, non-repeating decimals….}

Task: Graph Task: Graph yy = 2 = 2xx-1 and find -1 and find the domain & rangethe domain & range

• Once again use and x-y box. (y=mx+b)

• Fill in -4 for x.y=2(-4)-1y=-8-1y= -9

Do the same for the rest of the values chosen.

x y-4-2 0+2+4

-9

When you are finished, go to the next slide.

Graph the pointsGraph the points

Add a line

x y-4 -9-2 -5 0 -1+2 3+4 7

Name the domain and Name the domain and range.range.

Any number can be used as x or y.

Domain:{all real numbers}

Range:{all real numbers}

x y-4 -9-2 -5 0 -1+2 3+4 7

Graph y = | x-2 |Graph y = | x-2 |

• Start by using an x-y box with 0 and some negative and positive numbers for x.|-5-2| = |-7||-7| = 7

+7

x y-5-1 0+2+6+8

Graph y = | x-2 |Graph y = | x-2 |

• Show the graphed pairs.

• Fill in a few more values that work. x y

-5 7-1 3 0 2+2 0+6 4+8 6

IsIs y = | x-2 | y = | x-2 | a linear a linear equation?equation?

• You can begin to see that the values form a V when graphed, not a line.

Any real number can be used as x, but no negative numbers are used for y.

Domain:{all real numbers}Range:{all wholel

numbers}

How is y = - | x-2 | How is y = - | x-2 | different?different?

• All the y values are opposite the previous equation’s y-values. x y

-5 7-1 3 0 2+2 0+6 4+8 6

x y-5 -7-1 -3 0 -2+2 -0+6 -4+8 -6

Absolute Value Equations Absolute Value Equations with Inequalitieswith Inequalities

Key: Split the equation into two parts, a positive

and negative side.

Absolute ValueAbsolute Value

• To find Absolute value,– find the solution

inside the absolute value signs

– Make that value positive (+)

– Continue on with order of operations outside the signs

• Example:

7 | 2 3 | 7 5

Making Use of Absolute Making Use of Absolute ValueValue

• Adding a positive to a negative integer– Which has the higher

absolute value?– The positive or

negative sign of that number is in the answer.

– Now find the difference.

14 ( 27) - 13

Find the value: |x-2| =7Find the value: |x-2| =7

• This has two possible answers.

• There must be a handy pattern to use to find both.

• |+9-2| =7• |-5-2| =7

How to find the value: |x-2| How to find the value: |x-2| =7=7

• This problem should be done twice.

• Procedure:– Remove the absolute value signs– Solve for the positive answer.– Rewrite without absolute value

signs.– Solve for negative answer.

Procedure |x-2| =7Procedure |x-2| =7

• Remove absolute value signs.

x - 2 = 7• Solve for x

x +2 -2 = +2 + 7

x = 9

• Make 2nd equation’s answer negative.

x - 2 = -7• Solve for x

x +2 -2 = +2 - 7x = -5

Let’s take another look at a previous slide and see if the answers given were correct.

Find the value: |x-2| =7Find the value: |x-2| =7

• This has two possible answers.

• There must be a handy pattern to use to find both.

|+9-2| =7|-5-2| =7

x = -5 OR +9Give both answers.

Procedure for |Procedure for |x-10x-10| =4.5| =4.5

• Remove absolute value signs.

x - 10 = 4.5• Solve for x

x +10 -10 = +10 + 4.5

x = 14.5

• Make 2nd equation’s answer negative.

x - 10 = -4.5• Solve for x

x +10 -10 = +10 – 4.5

x = -5.5

x = -5.5 OR +14.5

Solve for | 2Solve for | 2x-14 x-14 | = 8| = 8

• Part One. 2x - 14 = 8

+14 +14 2x +0 = 22

x = +11

• Part Two. 2x - 14 = -8

+14 +14

2x +0 = 6

x = 3 x = +3 OR +11

2 22

2 2

x 2 6

2 2

x

Solve for |x - (-5)| Solve for |x - (-5)| 8 8

• Part One.

x + 5 8 -5 -5 x +0 3

x +3

• Switch the sign for the negative. Why?

x + 5 -8 -5 -5

x +0 -13

x -13 x -13 OR x

+3

Graph the solution forGraph the solution for |x - (-5)| |x - (-5)| 8 8

• You can rewrite the OR statement.

• Then graph.

• x -13 OR x +3

• -13 x +3

-6 -4 -2 0 +2 +4 +6

Graph the solution to the Graph the solution to the equation.equation.

-14 -12 -10 -8 -6 +4 -2 0 +2

13 3x

Solve for |x - 6| Solve for |x - 6| >> 5 5

• Part One.

x - 6 > 5 +6 +6 x +0 > 11

x > +11

• Switch the sign for the negative. Why?

x - 6 < -5 +6 +6

x +0 < +1

x < +1 x > +1 OR x <

+11

Graph the solution to Graph the solution to |x - 6| |x - 6| >> 5 5

-4 -2 0 2 4 6 8 10 12

x > +1 OR x < +11

Extras for presentationExtras for presentation

x y-4-2 0+2+4

-6 -4 -2 0 +2 +4 +6