Types of numbers Notation for intervals Inequalities Absolute value

Appendix A:Numbers, Inequalities, and

Absolute Values

Tom Lewis

Fall Semester2015

Types of numbers Notation for intervals Inequalities Absolute value

Outline

Types of numbers

Notation for intervals

Inequalities

Absolute value

Types of numbers Notation for intervals Inequalities Absolute value

A hierarchy of numbers

Whole numbers 1, 2, 3, . . .; these are the natural counting numbers.

Integers . . . ,−3,−2,−1, 0, 1, 2, 3, . . .; these are the wholenumbers together with their additive inverses and 0.This is a useful abstraction of the whole numbers.

Rational numbers −3/2, 111/53, .77777 · · · . These are numbersthat can be represented in the form p/q where p andq are integers. For example,

Irrational numbers√

2, π, 3√

5. These are the real numbers whichare not rational; their decimal expansions do notterminate nor do they repeat.

Types of numbers Notation for intervals Inequalities Absolute value

ProblemClassify each of the following numbers in its lowest possible numbertype.

• −6

• 3.134567676767 . . .

•185

45 + 2

5

•√

2

•√

4 +√

121 +√

3

Types of numbers Notation for intervals Inequalities Absolute value

The real numbers as a setThe real numbers are totally ordered. This means that if we aregiven any two distinct real numbers x and y , then they can becompared; thus, either

x > y or x < y .

We can visualize the real numbers therefore in a (familiar) numberline.

Types of numbers Notation for intervals Inequalities Absolute value

Subsets

• A subset of real numbers is any collection of real numbers.• By far the most important subsets of real numbers are theintervals.

• From a geometrical viewpoint, the intervals correspond toconnected segments within the real line.

• The endpoints may or not be contained in an interval. Aninterval can have infinite length.

Types of numbers Notation for intervals Inequalities Absolute value

We have two preferred notations: interval and set-builder notation.

ProblemSketch the intervals given by (−∞, 2] and {x : −3 6 x < 8}.

Types of numbers Notation for intervals Inequalities Absolute value

ProblemConsider the inequality:

2x + 5 < 4x + 8.

• What does it mean to solve the inequality?

• What kind of answer do we expect?

• How do we solve the inequality?

Types of numbers Notation for intervals Inequalities Absolute value

Solving inequalitiesWe can solve an inequality just as we would an equality with oneexception: multiplication by a negative number. If a < b andc < 0, then ca > cb.

Types of numbers Notation for intervals Inequalities Absolute value

Problem

1. Solve 3x + 12 < 6x − 18 for x .

2. Find the set of all x such that

3 − 5x > −3 and 3 − 5x < 18.

3. Find the set of all x such that

3x − 4 > 8 or 3x − 4 < −8.

4. Find the set of all t such that t2 + 3t > 28.

Types of numbers Notation for intervals Inequalities Absolute value

Definition (Absolute value)Given a real number x , the absolute value of x , denoted by |x |, isthe distance from the 0 (the origin) to x . We can think of |x | asthe length of the number x .

Types of numbers Notation for intervals Inequalities Absolute value

ProblemHow can you make a simple calculator return the absolute value ofan input?

Types of numbers Notation for intervals Inequalities Absolute value

Definition (Absolute value)We can formulate a precise definition of |x | as follows:

|x | =

{x if x > 0

−x if x < 0.

Types of numbers Notation for intervals Inequalities Absolute value

ProblemIn each case, describe the indicated set as an interval or union ofintervals:

• |x | 6 3.

• |x | > 2.

• |x | 6 −1.

Types of numbers Notation for intervals Inequalities Absolute value

Problem

1. Solve |x | = 8.

2. Solve |x | < 2 and |x | > 5.

3. Solve |x − 3| < 1. What is the geometric meaning of this set?

4. Find a and b such that the solution set of |x − a | < b is theinterval (−1, 7).

5. Solve |2 − 3x | > 8.

6. Solve the equation |x + 2| = |3x − 1|.

Types of numbers Notation for intervals Inequalities Absolute value

Some properties of absolute valueLet a and b be real numbers.

•√

a2 = |a |

• |ab| = |a ||b|

• |a/b| = |a |/|b|, provided that b 6= 0

• |a + b| 6 |a |+ |b| (triangle inequality)

Types of numbers Notation for intervals Inequalities Absolute value

ProblemSuppose that |x − 3| < .001 and |y − 5| < .002. Use the triangleinequality to give an upper bound for |(x + y) − 8|.