Special Equations - Absolute Value and Interval Notation Recall your Algebra 1 class where you learned about absolute value. Absolute value was always

Special Equations - Absolute Value and Interval Notation

Recall your Algebra 1 class where you learned about absolute value. Absolute value was always positive, because it measures the distance from zero. We can evaluate absolute value equations by substituting given values into the equation.



EXAMPLE : 2 when 52 Evaluate xx




1154522




1154522

EXAMPLE : 4,2 when 3 Evaluate baba




1154522

EXAMPLE : 4,2 when 3 Evaluate baba

224646423


You also learned how to solve absolute value equations. Remember, all absolute value equations have two POSSIBLE answers.



1064 xEXAMPLE :



1064 xEXAMPLE :

106410 x

STEPS :

1. Drop the Absolute value sign and set the equation equal to the negative answer on the other side.



1064 xEXAMPLE :

41

1644

66 6

106410

x

x

x

STEPS :

1. Drop the Absolute value sign and set the equation equal to the negative answer on the other side.

2. Solve the equation for both sides


Interval notation is a method for expressing a solution involving inequalities using brackets. We will graph the solution first on a number line which will help us determine the interval.



When graphing : < or > uses an open circle

≤ or ≥ uses a closed circle

Squeeze – when both inequality symbols point left. The solution is “squeezed” in between values

Gap – when both inequality symbols point right. The solution has a “gap” in its graph







Example : Solve the inequality and graph your solution.

136 x








13613

136

x

x

Again, drop the absolute value sign, repeat your inequality, and set your negative answer








719

66 6

13613

136

x

x

x

Solve like you would any equation…








719

66 6

13613

136

x

x

x

Graph your points…

-19 7








719

66 6

13613

136

x

x

x

< symbol so open circles…

-19 7








719

66 6

13613

136

x

x

x

Both symbols point left, so squeeze your answer in between the points…

-19 7







Example # 2 : Solve the inequality and graph your solution.

2824 x








224

2824

2824

x

x

x

Notice how I simplified inside first before begin solving…








224

2824

2824

x

x

x

31

622

4 4 4

2242

x

x

x

Drop the absolute value sign, repeat your inequality, and set your negative answer and solve…








224

2824

2824

x

x

x

31

622

4 4 4

2242

x

x

x

Graph your points…

1 3








224

2824

2824

x

x

x

31

622

4 4 4

2242

x

x

x

≥ symbol so closed circles…

1 3








224

2824

2824

x

x

x

31

622

4 4 4

2242

x

x

x

≥ symbols point right, so a gap is in your answer…

1 3


Interval notation – uses open and closed intervals to describe solution sets



If you use the graphed solution set, an interval is easy…

Open circle – round brackets ( a , b ) OR ( − ∞, a ) U ( b , ∞ )

Closed circle – square brackets [ a , b ] OR ( − ∞, a ] U [ b , ∞ )

In each case, a and b are our graphed points…





Closed circle – square brackets [ a , b ] OR [ − ∞, a ] U [ b , ∞ ]


- 4 11

Let’s say this is our graphed answer to an Absolute Value Inequality Equation





Closed circle – square brackets [ a , b ] OR [ − ∞, a ] U [ b , ∞ ]


( - 4 11 )

Let’s say this is our graphed answer to an Absolute Value Inequality Equation

To turn this answer into an interval, squeeze your numbers between round brackets ( open circles )

( - 4 , 11 )

,





Closed circle – square brackets [ a , b ] OR ( − ∞, a ] U [ b , ∞ )


( - ∞ , - 4 ) ( 11 , ∞ )

If this is our graphed answer to an Absolute Value Inequality Equation…

To turn this answer into an interval, use the OR template…a and b are edges and the graph extends to infinity on both sides…

** infinity will ALWAYS have rounded brackets !!!

U


Some problems will give you the given interval that satisfies an absolute value inequality. The problem will ask for the equation that generated that interval.



edgean todistance midpoint

edgean todistancemidpoint

, OR ,Given

x

x

baba

2 midpoint

ba

Distance to an edge is the distance from either “a” or “b” to the midpoint

rounded brackets

square brackets





, OR ,Given

x

x

baba2

midpoint ba


EXAMPLE : Express the given interval as an absolute value inequality

) 6 , 4 (





, OR ,Given

x

x

baba2

midpoint ba



) 6 , 4 (

1st – Find the midpoint 12

2

2

64

2

ba

1xAnswer :





, OR ,Given

x

x

baba2

midpoint ba



) 6 , 4 (

1st – Find the midpoint

2nd – How far away is the midpoint from either edge ?

12

2

2

64

2

ba

1xAnswer :





, OR ,Given

x

x

baba2

midpoint ba



) 6 , 4 (


2nd – How far away is the midpoint from either edge ?

12

2

2

64

2

ba

51 xAnswer :

516





, OR ,Given

x

x

baba2

midpoint ba


EXAMPLE # 2 : Express the given interval as an absolute value inequality

4 , 1

1st – Find the midpoint2

5

2

41

2

ba

2

5xAnswer :





, OR ,Given

x

x

baba2

midpoint ba


EXAMPLE # 2 : Express the given interval as an absolute value inequality

4 , 1


2nd – How far from the midpoint is an edge ?

2

5

2

41

2

ba

2

3

2

5xAnswer :

2

3

2

54

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Special Equations - Absolute Value and Interval Notation Recall your Algebra 1 class where you learned about absolute value. Absolute value was always