Preparing for University Calculus - Math and Computer Science

3.3 Inequalities and absolute valuesYou should be able to solve simple inequalities and perform algebraic operationswith them. In particular, you should know which operations reverse inequalitiesand which ones preserve them. You should understand interval notation, includingopen, closed, and half-open intervals, and intervals with limits at ∞. You shouldknow how to compute an absolute value, and to do simple algebra using the absolutevalue function.Examples:1. Solve 7 − 2x ≤ 4.3Solution: 7 − 2x ≤ 4 implies 7 − 2x ≤ 12 and −2x ≤ 5.3Thus x ≥− 5 2 ,orx ∈ [− 5 2 , ∞).2. Solve x 2 +3x − 10 ≤ 0.Solution:Factoring the left side, (x + 5)(x − 2) ≤ 0. The corresponding equation(x + 5)(x − 2) = 0 has solutions −5 and 2 which divide the real line into threeintervals: (−∞, −5), (−5, 2), (2, ∞).On each of these intervals we determine the sign of the factors as follows:Interval x +5 x − 2 (x + 5)(x − 2)(−∞, −5) − − +(−5, 2) + − −(2, ∞) + + +From the table we find that x 2 +3x − 10 ≤ 0 on the interval x ∈ [−5, 2].3. Solve |4x +5| > 9.Solution: |4x +5| > 9 is equivalent to4x +5> 9 or 4x +5< −94x >4 or 4x1 or x