Appendix A: Numbers, Inequalities, and Absolute...
Transcript of Appendix A: Numbers, Inequalities, and Absolute...
![Page 1: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/1.jpg)
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
![Page 2: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/2.jpg)
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
![Page 3: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/3.jpg)
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.
![Page 4: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/4.jpg)
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?
![Page 5: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/5.jpg)
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.
![Page 6: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/6.jpg)
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?
![Page 7: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/7.jpg)
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.
![Page 8: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/8.jpg)
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)
![Page 9: Appendix A: Numbers, Inequalities, and Absolute Valuesmath.furman.edu/~tlewis/math140/stewart/Appendix/appa.pdf · Types of numbersNotation for intervalsInequalitiesAbsolute value](https://reader033.fdocuments.net/reader033/viewer/2022052017/603065b2e0787a0d8b2a3fed/html5/thumbnails/9.jpg)
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|.