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lecture01 - Lecture 1 Precalculus Review 1.1 Real Line and...

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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.
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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 half-open interval the half-open 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 < 0 factor (2 x - 1)( x - 4) < 0 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:
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(a) 2 x - 1 < 0 and x - 4 > 0, or (b) 2 x - 1 > 0 and x - 4 < 0 In case (a), the first inequality solves to x < 1 2 , while the second solves to
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