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Unformatted text preview: Lecture 1  Precalculus Review 1.1 Real Line and Order When discussing order on the real number line, we use the following symbols: < – less than ≤ – less than or equal to > – greater than ≥ – greater than or equal to We use the following notation for intervals: x ∈ ( a,b ) ⇔ a < x < b open interval the open inverval ( a,b ) 3 h h a b h h The open circle at the ends of the interval indicates that the endpoint is not included. x ∈ [ a,b ] ⇔ a ≤ x ≤ b closed interval the closed inverval [ a,b ] 3 x x a b x x x ∈ [ a,b ) ⇔ a ≤ x < b halfopen interval the halfopen interval [ a,b ) 3 x h a b x h Examples 1. Solve 1 < x 3 < 2. solution: multiply by 3 ⇒  3 < x < 6 multiply by 1 ⇒ 3 > x > 6 ⇒  6 < x < 3 (Remember, when solving inequalities, if you multiply by a negative number you must flip the inequality.) Thus the set of all solutions is the open interval ( 6 , 3). 2. Solve 2 x 2 + 1 < 9 x 3. solution: 2 x 2 + 1 < 9 x 3 ⇒ 2 x 2 9 x + 4 < factor ⇒ (2 x 1)( x 4) < To have (2 x 1)( x 4) < 0, we have to have one of the factors being negative and one of the factors being positive. Thus we have the two cases: (a) 2 x 1 < 0 and x 4 > 0, or (b) 2 x 1 > 0 and x 4 < In case (a), the first inequality solves to...
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This note was uploaded on 01/07/2010 for the course MAT mat1300 taught by Professor Pieterhofstra during the Fall '09 term at University of Ottawa.
 Fall '09
 PIETERHOFSTRA
 Calculus

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