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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 positive, because it measures the distance from zero. We can evaluate absolute value equations by substituting given values into the equation.
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
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
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
EXAMPLE : 4,2 when 3 Evaluate baba
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
EXAMPLE : 4,2 when 3 Evaluate baba
224646423
Special Equations - Absolute Value and Interval Notation
You also learned how to solve absolute value equations. Remember, all absolute value equations have two POSSIBLE answers.
Special Equations - Absolute Value and Interval Notation
You also learned how to solve absolute value equations. Remember, all absolute value equations have two POSSIBLE answers.
1064 xEXAMPLE :
Special Equations - Absolute Value and Interval Notation
You also learned how to solve absolute value equations. Remember, all absolute value equations have two POSSIBLE answers.
1064 xEXAMPLE :
106410 x
STEPS :
1. Drop the Absolute value sign and set the equation equal to the negative answer on the other side.
Special Equations - Absolute Value and Interval Notation
You also learned how to solve absolute value equations. Remember, all absolute value equations have two POSSIBLE answers.
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
Special Equations - Absolute Value and Interval Notation
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.
Special Equations - Absolute Value and Interval Notation
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
Special Equations - Absolute Value and Interval Notation
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
Special Equations - Absolute Value and Interval Notation
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.
13613
136
x
x
Again, drop the absolute value sign, repeat your inequality, and set your negative answer
Special Equations - Absolute Value and Interval Notation
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.
719
66 6
13613
136
x
x
x
Solve like you would any equation…
Special Equations - Absolute Value and Interval Notation
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.
719
66 6
13613
136
x
x
x
Graph your points…
-19 7
Special Equations - Absolute Value and Interval Notation
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.
719
66 6
13613
136
x
x
x
< symbol so open circles…
-19 7
Special Equations - Absolute Value and Interval Notation
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.
719
66 6
13613
136
x
x
x
Both symbols point left, so squeeze your answer in between the points…
-19 7
Special Equations - Absolute Value and Interval Notation
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 # 2 : Solve the inequality and graph your solution.
2824 x
Special Equations - Absolute Value and Interval Notation
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 # 2 : Solve the inequality and graph your solution.
224
2824
2824
x
x
x
Notice how I simplified inside first before begin solving…
Special Equations - Absolute Value and Interval Notation
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 # 2 : Solve the inequality and graph your solution.
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…
Special Equations - Absolute Value and Interval Notation
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 # 2 : Solve the inequality and graph your solution.
224
2824
2824
x
x
x
31
622
4 4 4
2242
x
x
x
Graph your points…
1 3
Special Equations - Absolute Value and Interval Notation
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 # 2 : Solve the inequality and graph your solution.
224
2824
2824
x
x
x
31
622
4 4 4
2242
x
x
x
≥ symbol so closed circles…
1 3
Special Equations - Absolute Value and Interval Notation
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 # 2 : Solve the inequality and graph your solution.
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
Special Equations - Absolute Value and Interval Notation
Interval notation – uses open and closed intervals to describe solution sets
Special Equations - Absolute Value and Interval Notation
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…
Special Equations - Absolute Value and Interval Notation
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…
- 4 11
Let’s say this is our graphed answer to an Absolute Value Inequality Equation
Special Equations - Absolute Value and Interval Notation
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…
( - 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 )
,
Special Equations - Absolute Value and Interval Notation
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…
( - ∞ , - 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
Special Equations - Absolute Value and Interval Notation
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.
Special Equations - Absolute Value and Interval Notation
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
Special Equations - Absolute Value and Interval Notation
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
baba2
midpoint ba
Distance to an edge is the distance from either “a” or “b” to the midpoint
EXAMPLE : Express the given interval as an absolute value inequality
) 6 , 4 (
Special Equations - Absolute Value and Interval Notation
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
baba2
midpoint ba
Distance to an edge is the distance from either “a” or “b” to the midpoint
EXAMPLE : Express the given interval as an absolute value inequality
) 6 , 4 (
1st – Find the midpoint 12
2
2
64
2
ba
1xAnswer :
Special Equations - Absolute Value and Interval Notation
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
baba2
midpoint ba
Distance to an edge is the distance from either “a” or “b” to the midpoint
EXAMPLE : Express the given interval as an absolute value inequality
) 6 , 4 (
1st – Find the midpoint
2nd – How far away is the midpoint from either edge ?
12
2
2
64
2
ba
1xAnswer :
Special Equations - Absolute Value and Interval Notation
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
baba2
midpoint ba
Distance to an edge is the distance from either “a” or “b” to the midpoint
EXAMPLE : Express the given interval as an absolute value inequality
) 6 , 4 (
1st – Find the midpoint
2nd – How far away is the midpoint from either edge ?
12
2
2
64
2
ba
51 xAnswer :
516
Special Equations - Absolute Value and Interval Notation
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
baba2
midpoint ba
Distance to an edge is the distance from either “a” or “b” to the midpoint
EXAMPLE # 2 : Express the given interval as an absolute value inequality
4 , 1
1st – Find the midpoint2
5
2
41
2
ba
2
5xAnswer :
Special Equations - Absolute Value and Interval Notation
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
baba2
midpoint ba
Distance to an edge is the distance from either “a” or “b” to the midpoint
EXAMPLE # 2 : Express the given interval as an absolute value inequality
4 , 1
1st – Find the midpoint
2nd – How far from the midpoint is an edge ?
2
5
2
41
2
ba
2
3
2
5xAnswer :
2
3
2
54