2. CHAPTER R: Basic Concepts of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation, and Order of
Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

3. R.1 The Real-Number System
Identify various kinds of real numbers.
Use interval notation to write a set of numbers.
Identify the properties of real numbers.
Find the absolute value of a real number.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

4. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Rational Numbers
Numbers that can be expressed in the form p/q, where p and q are integers and q  0.
Decimal notation for rational numbers either terminates (ends) or repeats.
Examples:
a) b)
c) 9 d)
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

5. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Irrational Numbers
The real numbers that are not rational are irrational numbers.
Decimal notation for irrational numbers neither terminates nor repeats.
Examples:
a) … b)
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

6. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Interval Notation
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

7. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Write interval notation for each set and graph the set.
a) {x|4 < x < 5}
Solution: {x|4 < x < 5} = (4, 5)
b) {x|x ≥ 1.7}
Solution: {x| x ≥ 1.7} = [1.7, ∞)
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

8. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Write interval notation for each set and graph the set.
c) {x|5 < x ≤ ‒2}
Solution: {x| 5 < x ≤ ‒2} = (5, ‒2]
d) {x|x < }
Solution: {x| x < } = (∞, )
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

9. Properties of the Real Numbers
Commutative properties of addition and multiplication
a + b = b + a and ab = ba
Associative properties of addition and multiplication
a + (b + c) = (a + b) + c and
a(bc) = (ab)c
Additive identity property
a + 0 = 0 + a = a
Additive inverse property
a + a = a + (a) = 0
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

10. More Properties of Real Numbers
Multiplicative identity property
a • 1 = 1 • a = a
Multiplicative inverse property
Distributive property
a(b + c) = ab + ac
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

11. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
State the property being illustrated in each sentence.
a) (7)(6) = (6)(7) Commutative Property of Multiplication
b) 3d + 3c = 3(d + c) Distributive
c) (3 + y) + x = 3 + (y + x) Associative Property of Addition
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

12. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Absolute Value
The absolute value of a number a, denoted |a|, is its distance from 0 on the number line.
Example:
Simplify.
|6| = 6
|19| = 19
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

13. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Absolute Value
For any real number a,
When a is nonnegative, the absolute value of a is a.
When a is negative, the absolute value of a is the opposite of a.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

14. Distance Between Two Points on the Number Line
For any real numbers a and b, the distance between a and b is |a  b|, or equivalently |b  a|.
Example:
Find the distance between 2 and 3.
Solution: The distance is |2  3| = |5| = 5,
or |3 (2)| = |3 + 2| = |5| = 5.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley