CHAPTER

0Preliminaries

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0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE0.2 LINES AND FUNCTIONS0.3 GRAPHING CALCULATORS AND COMPUTER

ALGEBRA SYSTEMS0.4 TRIGONOMETRIC FUNCTIONS0.5 TRANSFORMATIONS OF FUNCTIONS

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Real Number System and Inequalities

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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 −7/3are 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.

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Real Number System and Inequalities

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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.

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Real Number System and Inequalities

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We picture the real numbers arranged along the number line (the real line). The set of real numbers is denoted by the symbol .

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Real Number System and Inequalities

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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, .

THEOREM

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.1

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EXAMPLE

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.1 Solving a Linear Inequality

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EXAMPLE

Solution

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.1 Solving a Linear Inequality

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EXAMPLE

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.2 Solving a Two-Sided Inequality

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EXAMPLE

Solution

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.2 Solving a Two-Sided Inequality

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EXAMPLE

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.3 Solving an Inequality Involving a Fraction

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EXAMPLE

Solution

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.3 Solving an Inequality Involving a Fraction

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EXAMPLE

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.4 Solving a Quadratic Inequality

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Solve the quadratic inequality

EXAMPLE

Solution

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.4 Solving a Quadratic Inequality

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DEFINITION

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE1.1

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0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Real Number System and Inequalities

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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| asthe distance between a and b is particularly useful for solving inequalities involving absolute values.

EXAMPLE

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.7 Solving Inequalities

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Solve the inequality |x − 2|< 5.

EXAMPLE

Solution

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.7 Solving Inequalities

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0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Cartesian Plane

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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 tworeal number lines drawn at right angles.

The horizontal line is called thex-axis and the vertical line iscalled the y-axis.

The point where the axes cross is calledthe origin, which represents the ordered pair (0, 0).

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Cartesian Plane

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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).

THEOREM

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.1

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The distance between the points (x1, y1) and (x2, y2) in the Cartesian plane is given by

EXAMPLE

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.9 Using the Distance Formula

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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.

EXAMPLE

Solution

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

1.9 Using the Distance Formula

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The triangle is not a right triangle.