Outline 1 Real Numbers 2 Inequalities and Absolute Values 3 The Rectangular Coordinate System 4 Graphs of Equations 5 Functions and Their Graphs 6 Operations on Functions 7 Trigonometri Functions Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 17 / 107 0.2 Inequalities and Absolute Values Examples of equations 3x 17 = 6, x2 x 6=0 x2 x 6 Examples of inequalities 3x 17 < 6, 0 Solving equations is one of the traditional tasks of mathematics. But of almost equal significance in calculus is the notion of solving an inequality. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 18 / 107 0.2 Inequalities and Absolute Values Examples of equations 3x 17 = 6, x2 x 6=0 x2 x 6 Examples of inequalities 3x 17 < 6, 0 Solving equations is one of the traditional tasks of mathematics. But of almost equal significance in calculus is the notion of solving an inequality. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 18 / 107 0.2 Inequalities and Absolute Values Examples of equations 3x 17 = 6, x2 x 6=0 x2 x 6 Examples of inequalities 3x 17 < 6, 0 Solving equations is one of the traditional tasks of mathematics. But of almost equal significance in calculus is the notion of solving an inequality. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 18 / 107 Solutions of Inequalities To solve an inequality, for example, x2 x 6 0, is to find the set of all real numbers that make the inequality true. The solution set of an inequality is usually an entire interval of numbers or, in some cases, the union of such intervals. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 19 / 107 Solutions of Inequalities To solve an inequality, for example, x2 x 6 0, is to find the set of all real numbers that make the inequality true. The solution set of an inequality is usually an entire interval of numbers or, in some cases, the union of such intervals. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 19 / 107 Open Interval (a, b) Several kinds of intervals will arise in our work and we introduce special terminology and notation for them. The inequality a < x < b, which is actually two inequalities, a < x and x < b, describes the open interval consisting of all numbers between a and b, not including the end points a and b. We denote this interval by the symbol (a, b) (Figure 1). (round bracket) Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 20 / 107 Open Interval (a, b) Several kinds of intervals will arise in our work and we introduce special terminology and notation for them. The inequality a < x < b, which is actually two inequalities, a < x and x < b, describes the open interval consisting of all numbers between a and b, not including the end points a and b. We denote this interval by the symbol (a, b) (Figure 1). (round bracket) Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 20 / 107 Open Interval (a, b) Several kinds of intervals will arise in our work and we introduce special terminology and notation for them. The inequality a < x < b, which is actually two inequalities, a < x and x < b, describes the open interval consisting of all numbers between a and b, not including the end points a and b. We denote this interval by the symbol (a, b) (Figure 1). (round bracket) Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 20 / 107 Closed Interval [a, b] The inequality a x b describes the corresponding closed interval, which does include the end points a and b. This interval is denoted by [a, b] (Figure 2). (square bracket) Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 21 / 107 Possible Intervals The table below indicates the wide variety of possibilities. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 22 / 107 Solving Inequalities We may perform certain operations on both sides of an inequality without changing its solution set. In particular, 1. We may add the same number to both sides of an inequality. 2. We may multiply both sides of an inequality by the same positive number. 3. We may multiply both sides by the same negative number, but then we must reverse the direction of the inequality sign. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 23 / 107 Solving Inequalities We may perform certain operations on both sides of an inequality without changing its solution set. In particular, 1. We may add the same number to both sides of an inequality. 2. We may multiply both sides of an inequality by the same positive number. 3. We may multiply both sides by the same negative number, but then we must reverse the direction of the inequality sign. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 23 / 107 Example 1 Solve the inequality 2x set. Higher Mathematics G1 (Dept. Math) 7 < 4x 2 and show the graph of its solution Chapter 0 Preliminaries 2019-2020-01 24 / 107 Example 1 Solve the inequality 2x set. Higher Mathematics G1 (Dept. Math) 7 < 4x 2 and show the graph of its solution Chapter 0 Preliminaries 2019-2020-01 24 / 107 Example 2 Solve the inequality set. Higher Mathematics G1 (Dept. Math) 5 2x + 6 < 4 and show the graph of its solution Chapter 0 Preliminaries 2019-2020-01 25 / 107 Example 2 Solve the inequality set. Higher Mathematics G1 (Dept. Math) 5 2x + 6 < 4 and show the graph of its solution Chapter 0 Preliminaries 2019-2020-01 25 / 107 Split Points Before tackling a quadratic inequality, we point out that a linear factor of the form x a is positive for x > a and negative for x < a. It follows that the product (x a)(x b) can change from being positive to negative, or vice versa, only at a or b. These points, where a factor ((x split points. a) or (x b)) is zero, are called They are the keys to determining the solution sets of quadratic and other more complicated inequalities. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 26 / 107 Split Points Before tackling a quadratic inequality, we point out that a linear factor of the form x a is positive for x > a and negative for x < a. It follows that the product (x a)(x b) can change from being positive to negative, or vice versa, only at a or b. These points, where a factor ((x split points. a) or (x b)) is zero, are called They are the keys to determining the solution sets of quadratic and other more complicated inequalities. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 26 / 107 Split Points Before tackling a quadratic inequality, we point out that a linear factor of the form x a is positive for x > a and negative for x < a. It follows that the product (x a)(x b) can change from being positive to negative, or vice versa, only at a or b. These points, where a factor ((x split points. a) or (x b)) is zero, are called They are the keys to determining the solution sets of quadratic and other more complicated inequalities. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 26 / 107 Split Points Before tackling a quadratic inequality, we point out that a linear factor of the form x a is positive for x > a and negative for x < a. It follows that the product (x a)(x b) can change from being positive to negative, or vice versa, only at a or b. These points, where a factor ((x split points. a) or (x b)) is zero, are called They are the keys to determining the solution sets of quadratic and other more complicated inequalities. Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 26 / 107 Example 3 Solve the quadratic inequality x 2 Higher Mathematics G1 (Dept. Math) x < 6. Chapter 0 Preliminaries 2019-2020-01 27 / 107 Example 3 Solve the quadratic inequality x 2 Higher Mathematics G1 (Dept. Math) x < 6. Chapter 0 Preliminaries 2019-2020-01 27 / 107 Example 3 Solve the quadratic inequality x 2 x < 6. Besides factorization, we can also use quadratic formula to find the split points. p b ± b2 4ac x= 2a Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 27 / 107 Example 3 (Con’t) From (x 3)(x + 2) < 0, Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 28 / 107 Example 4 Solve 3x 2 x 2>0 Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 29 / 107 Example 4 Solve 3x 2 x 2>0 Higher Mathematics G1 (Dept. Math) Chapter 0 Preliminaries 2019-2020-01 29 / 107
