Special Equations - Absolute Value and Interval Notation

1. 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.

2. 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 :

3. 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 :

4. 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 : EXAMPLE :

5. 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 : EXAMPLE :

6. 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.

7. 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. EXAMPLE :

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

9. 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. EXAMPLE : • STEPS : • Drop the Absolute value sign and set the equation equal to the negative answer on the other side. • Solve the equation for both sides

10. 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.

11. 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

12. 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.

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

14. 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. Solve like you would any equation…

15. 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. 7 -19 Graph your points…

16. 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. 7 -19 < symbol so open circles…

17. 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. 7 -19 Both symbols point left, so squeeze your answer in between the points…

18. 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.

19. 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. Notice how I simplified inside first before begin solving…

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

21. 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. 3 1 Graph your points…

22. 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. 3 1 ≥ symbol so closed circles…

23. 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. 3 1 ≥ symbols point right, so a gap is in your answer…

24. Special Equations - Absolute Value and Interval Notation Interval notation – uses open and closed intervals to describe solution sets

25. 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…

26. 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

27. 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 )

28. 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 , ∞ ) U 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 !!!

29. 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.

30. 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. rounded brackets square brackets Distance to an edge is the distance from either “a” or “b” to the midpoint

31. 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. 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

32. 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. 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 Answer : 1st – Find the midpoint

33. 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. 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 Answer : 1st – Find the midpoint 2nd – How far away is the midpoint from either edge ?

34. 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. 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 Answer : 1st – Find the midpoint 2nd – How far away is the midpoint from either edge ?

35. 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. 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 Answer : 1st – Find the midpoint

36. 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. 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 Answer : 1st – Find the midpoint 2nd – How far from the midpoint is an edge ?