2. CHAPTER 0 Preliminaries Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 0.1THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2LINES AND FUNCTIONS 0.3GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.4TRIGONOMETRIC FUNCTIONS 0.5TRANSFORMATIONS OF FUNCTIONS

3. 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Real Number System and Inequalities Slide 3 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The set of integers consists of the whole numbers and their additive inverses: 0, ±1,±2,±3,.... A rational number is any number of the form p/q, where p and q are integers and q ≠ 0. For example, 2/3 and −7/3 are rational numbers. Notice that every integer n is also a rational number, since we can write it as the quotient of two integers: n = n/1.

4. 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Real Number System and Inequalities Slide 4 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The irrational numbers are all those real numbers that cannot be written in the form p/q, where p and q are integers. Recall that rational numbers have decimal expansions that either terminate or repeat. By contrast, irrational numbers have decimal expansions that do not repeat or terminate.

5. 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Real Number System and Inequalities Slide 5 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We picture the real numbers arranged along the number line (the real line). The set of real numbers is denoted by the symbol.

6. 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Real Number System and Inequalities Slide 6 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For real numbers a and b, where a < b, we define the closed interval [a, b] to be the set of numbers between a and b, including a and b (the endpoints). That is,. Similarly, the open interval (a, b) is the set of numbers between a and b, but not including the endpoints a and b, that is,.

7. THEOREM 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.1 Slide 7 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

8. EXAMPLE 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.1Solving a Linear Inequality Slide 8 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

9. EXAMPLE Solution 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.1Solving a Linear Inequality Slide 9 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

10. EXAMPLE 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.2Solving a Two-Sided Inequality Slide 10 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

11. EXAMPLE Solution 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.2Solving a Two-Sided Inequality Slide 11 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

12. EXAMPLE 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.3Solving an Inequality Involving a Fraction Slide 12 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13. EXAMPLE Solution 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.3Solving an Inequality Involving a Fraction Slide 13 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14. EXAMPLE 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.4Solving a Quadratic Inequality Slide 14 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Solve the quadratic inequality

15. EXAMPLE Solution 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.4Solving a Quadratic Inequality Slide 14 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

16. DEFINITION 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.1 Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

17. 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Real Number System and Inequalities Slide 17 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Notice that for any real numbers a and b, |a · b| = |a| · |b|, although |a + b| ≠ |a| + |b|, in general. However, it is always true that |a + b| ≤ |a| + |b|. This is referred to as the triangle inequality. The interpretation of |a − b| as the distance between a and b is particularly useful for solving inequalities involving absolute values.

18. EXAMPLE 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.7Solving Inequalities Slide 18 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Solve the inequality |x − 2|< 5.

19. EXAMPLE Solution 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.7Solving Inequalities Slide 19 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

20. 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Cartesian Plane Slide 20 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For any two real numbers x and y we visualize the ordered pair (x, y) as a point in two dimensions. The Cartesian plane is a plane with two real number lines drawn at right angles. The horizontal line is called the x-axis and the vertical line is called the y-axis. The point where the axes cross is called the origin, which represents the ordered pair (0, 0).

21. 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Cartesian Plane Slide 21 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To represent the ordered pair (1, 2), start at the origin, move 1 unit to the right and 2 units up and mark the point (1, 2).

22. THEOREM 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.1 Slide 22 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The distance between the points (x 1, y 1 ) and (x 2, y 2 ) in the Cartesian plane is given by

23. EXAMPLE 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.9Using the Distance Formula Slide 23 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the distances between each pair of points (1, 2), (3, 4) and (2, 6). Use the distances to determine if the points form the vertices of a right triangle.

24. EXAMPLE Solution 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 1.9Using the Distance Formula Slide 24 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The triangle is not a right triangle.