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Algebra – 3.3 Distance and Absolute Value DISTANCE • What’s the distance between these points? • • • • (-1, -3) and (4, -3) (-1, -3) and (-1, 9) (2, 1) and (-3, 1) (2, 1) and (2, -16) DISTANCE • Can distance be negative? • How do these compare: – Distance from 1 to 4 – Distance from 4 to 1 • The easiest way to calculate distance on a number line is to do subtraction • But which number do you subtract? – “What is the distance between a and b?” – If a > b, then a – b. If b > a, then b – a. • I don’t like that! It’s wordy! • There’s a better way. Absolute Value DEFINITION: If x and y are numbers, then “the absolute value of (x-y)”, written |x – y| is the distance between x and y. Example: |8 – 3| = 5 and |3 – 8| = 5 since 8 and 3 are 5 units apart • Ex: |1 – (-5)| = • Ex: |-6 – 0| = • Ex: |(-2) – 6| = • Ex: |4 – 10| = • Ex: |(-3) – (-5)| = Special case • What does THIS mean? |8 + 10| |8| Theorem The absolute value or a number x (written: |x|) is its distance from 0 on the number line. PROOF: |x – 0| is the distance between x and 0. x–0=x By substitution, |x – 0| = |x| Therefore, |x| is the distance between x and 0. Algebraic representation |x| = x, if x≥0 -x, if x<0