Directed Distance & Absolute Value Objective: To be able to find directed distances and solve absolute value inequalities. TS: Making Decisions after Reflection.

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Presentation transcript:

Directed Distance & Absolute Value Objective: To be able to find directed distances and solve absolute value inequalities. TS: Making Decisions after Reflection and Review Warm Up: Solve the following absolute value equation. |2x – 3| = 13 The inside could have been+ or – 13 so need to solve both!

Distance Between Two Points. What is the distance between the two values of 10 and 2? What is the distance between the two values of -102 and 80? So the distance between two points x 1 and x 2 is |x 1 – x 2 | or |x 2 – x 1 | 8 182

Directed Distance The directed distance from a to b is b – a. Ex: Find the directed distance from 5 to – The directed distance from b to a is a – b. Ex: Find the directed distance from -10 to 5 5 – (-10) Had to go down, so Had to go up, so +15

Midpoint The midpoint between to values is a + b 2 Ex: Find the midpoint of the interval [1, 10]

Absolute Value Not true Is this statement true?

Absolute Value Think of absolute value as measuring a distance.

Absolute Value Absolute Value: The distance a number is from zero on a number line. It is always positive or zero.

Absolute Value ( ) The < sign indicates that the value is center around 0 and no more than 3 away.

Absolute Value ( ) The < sign indicates that the value is centered around 2 and no more than 3 away. NOTICE: 2 is the midpoint of -1 and 5. Now the subtraction of 2 has “translated” our center to 2.

Absolute Value ] [ The > sign indicates that the value is diverging from points on either side of 0.

Absolute Value ] [ The > sign indicates that the value is diverging from points on either side of -3. NOTICE: -3 is the midpoint of -4 and -1. Now the subtraction of -3 has “translated” our center to -3.

Writing an Absolute Value 1)Write an absolute value inequality for the below intervals: (-∞, - 4]U[4, ∞) (-5, 5) (- ∞, 2)U(5, ∞) [-10, 20] Ans: |x|≥4 Ans: |x|<5 Ans: |x – 3.5|>1.5 Ans: |x – 5|≤15

Absolute Value What does this statement mean? ( )

Absolute Value ] [ What does this statement mean?

Absolute Value Think of what we just saw. This picture would have two pieces, since the “distance” is greater.

Absolute Value Think of what we just saw. This picture would have one piece between two numbers, since the “distance” is smaller.

You Try Solve the following inequalities: 1) |2x|< 6 2) |3x+1|≥4 3) |25 – x|>20 Ans: (-3, 3) Ans: (-∞,-5/3] U [1,∞) Ans: (-∞,5) U (45,∞)

Conclusion Absolute value is the distance a number is from zero on a number line. Two equations are necessary to solve an absolute value equation. Two inequalities are necessary to solve an absolute value inequality.